Failed list 😭😭😭
Initialize a table for testing..
-- A. Create database and table
CREATE TABLE LOL_PRO (
ID INT PRIMARY KEY AUTO_INCREMENT NOT NULL,
LEAGUE VARCHAR(32) NOT NULL,
TEAM VARCHAR(32) NOT NULL,
RANK INT NOT NULL
);-- B. Initialize table data
INSERT INTO LOL_PRO(LEAGUE, TEAM, RANK) VALUES ("LPL", "Top Esport", 1);
INSERT INTO LOL_PRO(LEAGUE, TEAM, RANK) VALUES ("LPL", "JD Gaming", 2);
INSERT INTO LOL_PRO(LEAGUE, TEAM, RANK) VALUES ("LPL", "Invictus Gaming", 3);
INSERT INTO LOL_PRO(LEAGUE, TEAM, RANK) VALUES ("LPL", "Suning", 4);
INSERT INTO LOL_PRO(LEAGUE, TEAM, RANK) VALUES ("LPL", "Victory Five", 5);
INSERT INTO LOL_PRO(LEAGUE, TEAM, RANK) VALUES ("LCK", "DRX", 1);
INSERT INTO LOL_PRO(LEAGUE, TEAM, RANK) VALUES ("LCK", "DAMWON Gaming", 2);
INSERT INTO LOL_PRO(LEAGUE, TEAM, RANK) VALUES ("LCK", "Gen.G", 3);
INSERT INTO LOL_PRO(LEAGUE, TEAM, RANK) VALUES ("LCK", "T1", 4);
INSERT INTO LOL_PRO(LEAGUE, TEAM, RANK) VALUES ("LCK", "Afreeca Freecs", 5);
INSERT INTO LOL_PRO(LEAGUE, TEAM, RANK) VALUES ("LEC", "MAD Lions", 1);
INSERT INTO LOL_PRO(LEAGUE, TEAM, RANK) VALUES ("LEC", "Rouge", 2);
INSERT INTO LOL_PRO(LEAGUE, TEAM, RANK) VALUES ("LEC", "Fnatic", 3);
INSERT INTO LOL_PRO(LEAGUE, TEAM, RANK) VALUES ("LEC", "SK Gaming", 4);
INSERT INTO LOL_PRO(LEAGUE, TEAM, RANK) VALUES ("LEC", "Excel Exports", 5);+----+--------+-----------------+------+
| ID | LEAGUE | TEAM | RANK |
+----+--------+-----------------+------+
| 1 | LPL | Top Esport | 1 |
| 2 | LPL | JD Gaming | 2 |
| 3 | LPL | Invictus Gaming | 3 |
| 4 | LPL | Suning | 4 |
| 5 | LPL | Victory Five | 5 |
| 6 | LCK | DRX | 1 |
| 7 | LCK | DAMWON Gaming | 2 |
| 8 | LCK | Gen.G | 3 |
| 9 | LCK | T1 | 4 |
| 10 | LCK | Afreeca Freecs | 5 |
| 11 | LEC | MAD Lions | 1 |
| 12 | LEC | Rouge | 2 |
| 13 | LEC | Fnatic | 3 |
| 14 | LEC | SK Gaming | 4 |
| 15 | LEC | Excel Exports | 5 |
+----+--------+-----------------+------+
-- C. Best Rank of each League
SELECT LEAGUE, MIN(RANK) AS BEST
FROM LOL_PRO
GROUP BY LEAGUE;+--------+--------+
| LEAGUE | RANK_1 |
+--------+--------+
| LCK | 1 |
| LEC | 1 |
| LPL | 1 |
+--------+--------+
-- D. Worst Rank of each League
SELECT LEAGUE, MAX(RANK) AS WORST
FROM LOL_PRO
GROUP BY LEAGUE;+--------+-------+
| LEAGUE | WORST |
+--------+-------+
| LCK | 5 |
| LEC | 5 |
| LPL | 5 |
+--------+-------+
-- E. Best Rank Team of each League
SELECT L.LEAGUE, L.TEAM
FROM LOL_PRO AS L
INNER JOIN(
SELECT LEAGUE, MIN(RANK) AS BEST_RANK
FROM LOL_PRO
GROUP BY LEAGUE
) AS L_BEST
ON L.LEAGUE = L_BEST.LEAGUE
WHERE L.RANK = L_BEST.BEST_RANK;+--------+------------+
| LEAGUE | TEAM |
+--------+------------+
| LPL | Top Esport |
| LCK | DRX |
| LEC | MAD Lions |
+--------+------------+
-- F. Worst Rank Team of each League
SELECT L.LEAGUE, L.TEAM
FROM LOL_PRO AS L
INNER JOIN(
SELECT LEAGUE, MAX(RANK) AS WORST_RANK
FROM LOL_PRO
GROUP BY LEAGUE
) AS L_WORST
ON L.LEAGUE = L_WORST.LEAGUE
WHERE L.RANK = L_WORST.WORST_RANK;+--------+----------------+
| LEAGUE | TEAM |
+--------+----------------+
| LPL | Victory Five |
| LCK | Afreeca Freecs |
| LEC | Excel Exports |
+--------+----------------+
-- G. Join two tables
SELECT BRT.LEAGUE, BRT.TEAM AS "Top ranking", WRT.TEAM AS "Bottom ranking"
FROM (
SELECT L.LEAGUE, L.TEAM
FROM LOL_PRO AS L
INNER JOIN(
SELECT LEAGUE, MIN(RANK) AS BEST_RANK
FROM LOL_PRO
GROUP BY LEAGUE
) AS L_BEST
ON L.LEAGUE = L_BEST.LEAGUE
WHERE L.RANK = L_BEST.BEST_RANK
) AS BRT
LEFT JOIN(
SELECT L.LEAGUE, L.TEAM
FROM LOL_PRO AS L
INNER JOIN(
SELECT LEAGUE, MAX(RANK) AS WORST_RANK
FROM LOL_PRO
GROUP BY LEAGUE
) AS L_WORST
ON L.LEAGUE = L_WORST.LEAGUE
WHERE L.RANK = L_WORST.WORST_RANK
) AS WRT
ON BRT.LEAGUE = WRT.LEAGUE
ORDER BY BRT.LEAGUE;+--------+-------------+----------------+
| LEAGUE | Top ranking | Bottom ranking |
+--------+-------------+----------------+
| LCK | DRX | Afreeca Freecs |
| LEC | MAD Lions | Excel Exports |
| LPL | Top Esport | Victory Five |
+--------+-------------+----------------+
Described about the problem in the comments of this code
import random
from collections import defaultdict
"""
Number matching problem
- All integers between 1 to R
- The numbers chosen should never be duplicated! (R >= N)
- Target numbers T (size N) -> [t1,t2,t3,...t_n-1, t_n]
- Chosen numbers C (size N-M) [c1,c2,...,c_n-m-1, c_n-m] with empty numbers (size M) [e1,e2,...,e_m-1,e_m]
- N > M
- Q. What is the maximum matching count for a given selected numbers?
- Q. What is the minimum matching count for a given selected numbers?
"""
def load_random_inputs(R=100,N=10,M=5):
assert R > N and N > M
numbers = list(range(1,R+1))
random.shuffle(numbers)
target_numbers = numbers[:N]
random.shuffle(numbers)
chosen_numbers = numbers[:N-M] + [0] * M
return target_numbers, chosen_numbers
"""
Just count matchings without empty numbers
"""
def search_minimum_matching(target_numbers, chosen_numbers):
matching_count = 0
for c in chosen_numbers:
if c in target_numbers:
matching_count += 1
return matching_count
"""
Maximum matching is a simple counting with empty numbers using target numbers not chosen
"""
def search_maximum_matching(target_numbers, chosen_numbers, M):
matching_count = 0
possible_numbers = []
for t in target_numbers:
if t not in chosen_numbers:
possible_numbers.append(t)
min_count = search_minimum_matching(target_numbers, chosen_numbers)
max_count = min_count + min(len(possible_numbers), M)
return max_countLet's run this code...
"""
>> R: 100 / all possible numbers are between 1 to 100
>> N: 10 / we will choose 10 numbers in R without duplicate
>> M: 5 / 5 numbers (N-M) are fixed and others (M) are empty
"""
R, N, M = 100, 10, 5
T, C = load_random_inputs(R,N,M)
minimum_matchings = search_minimum_matching(T,C)
maximum_matchings = search_maximum_matching(T,C,M)
print("> Target numbers : {}".format(T))
print("> Chosen numbers : {}".format(C))
print("> Minimum matchs : {}".format(minimum_matchings))
print("> Maximum matchs : {}".format(maximum_matchings))Output is ...
> Target numbers : [70, 69, 5, 49, 59, 68, 74, 100, 89, 35]
> Chosen numbers : [51, 4, 97, 18, 55, 0, 0, 0, 0, 0]
> Minimum matchs : 0
> Maximum matchs : 5def __any_single_list_about_N_dices_with_sum_S(N,S):
N = N # number of possible dices
D = 6-1 # maximum of dice
S = S # sum of target
R = S - N # at least 1 x 10 values are fixed
if R <= 0 : return [], 0, "Failed"
if R > (N*D) : return [], 0, "Failed"
cur_sum = 0
dices = []
initial_dices = [1] * N
for i in range(N):
if cur_sum + D < R:
cur_sum += D
dices.append(D)
elif cur_sum + D == R:
dices.append(D)
break
else: # cur_sum + D > R
remain_sum = R - cur_sum
if remain_sum > D:
# impossible case!
dices.append(-1)
return [], 0, "Failed"
else:
# remain_sum < D
# D <- remain_sum
dices.append(remain_sum)
break
dices += [0] * (N - len(dices))
final_dices = [i+d for i,d in zip(initial_dices, dices)]
return final_dices, sum(final_dices), "Success"Run this method...
print(__any_single_list_about_N_dices_with_sum_S(N=10, S=59))Output
([6, 6, 6, 6, 6, 6, 6, 6, 6, 5], 59, 'Success')def permutations_with_sum_M(M, logging=False):
possibles = []
def multi_loop(K, vals):
if K < 1:
possibles.append((vals, sum(vals)))
return
for i in range(1,K+1):
multi_loop(K-i, vals + [i])
multi_loop(K=M, vals=[])
print("- number of permutations with sum M({}) : {}".format(M,len(possibles)))
if logging:
print(possibles)Run this method..
permutations_with_sum_M(M=0, logging=True)
permutations_with_sum_M(M=6, logging=True)
permutations_with_sum_M(M=8, logging=True)Output
- number of permutations with sum M(0) : 1
[([], 0)]
- number of permutations with sum M(6) : 32
[([1, 1, 1, 1, 1, 1], 6), ([1, 1, 1, 1, 2], 6), ([1, 1, 1, 2, 1], 6), ([1, 1, 1, 3], 6), ([1, 1, 2, 1, 1], 6), ([1, 1, 2, 2], 6), ([1, 1, 3, 1], 6), ([1, 1, 4], 6), ([1, 2, 1, 1, 1], 6), ([1, 2, 1, 2], 6), ([1, 2, 2, 1], 6), ([1, 2, 3], 6), ([1, 3, 1, 1], 6), ([1, 3, 2], 6), ([1, 4, 1], 6), ([1, 5], 6), ([2, 1, 1, 1, 1], 6), ([2, 1, 1, 2], 6), ([2, 1, 2, 1], 6), ([2, 1, 3], 6), ([2, 2, 1, 1], 6), ([2, 2, 2], 6), ([2, 3, 1], 6), ([2, 4], 6), ([3, 1, 1, 1], 6), ([3, 1, 2], 6), ([3, 2, 1], 6), ([3, 3], 6), ([4, 1, 1], 6), ([4, 2], 6), ([5, 1], 6), ([6], 6)]
- number of permutations with sum M(8) : 128
[([1, 1, 1, 1, 1, 1, 1, 1], 8), ([1, 1, 1, 1, 1, 1, 2], 8), ([1, 1, 1, 1, 1, 2, 1], 8), ([1, 1, 1, 1, 1, 3], 8), ([1, 1, 1, 1, 2, 1, 1], 8), ([1, 1, 1, 1, 2, 2], 8), ([1, 1, 1, 1, 3, 1], 8), ([1, 1, 1, 1, 4], 8), ([1, 1, 1, 2, 1, 1, 1], 8), ([1, 1, 1, 2, 1, 2], 8), ([1, 1, 1, 2, 2, 1], 8), ([1, 1, 1, 2, 3], 8), ([1, 1, 1, 3, 1, 1], 8), ([1, 1, 1, 3, 2], 8), ([1, 1, 1, 4, 1], 8), ([1, 1, 1, 5], 8), ([1, 1, 2, 1, 1, 1, 1], 8), ([1, 1, 2, 1, 1, 2], 8), ([1, 1, 2, 1, 2, 1], 8), ([1, 1, 2, 1, 3], 8), ([1, 1, 2, 2, 1, 1], 8), ([1, 1, 2, 2, 2], 8), ([1, 1, 2, 3, 1], 8), ([1, 1, 2, 4], 8), ([1, 1, 3, 1, 1, 1], 8), ([1, 1, 3, 1, 2], 8), ([1, 1, 3, 2, 1], 8), ([1, 1, 3, 3], 8), ([1, 1, 4, 1, 1], 8), ([1, 1, 4, 2], 8), ([1, 1, 5, 1], 8), ([1, 1, 6], 8), ([1, 2, 1, 1, 1, 1, 1], 8), ([1, 2, 1, 1, 1, 2], 8), ([1, 2, 1, 1, 2, 1], 8), ([1, 2, 1, 1, 3], 8), ([1, 2, 1, 2, 1, 1], 8), ([1, 2, 1, 2, 2], 8), ([1, 2, 1, 3, 1], 8), ([1, 2, 1, 4], 8), ([1, 2, 2, 1, 1, 1], 8), ([1, 2, 2, 1, 2], 8), ([1, 2, 2, 2, 1], 8), ([1, 2, 2, 3], 8), ([1, 2, 3, 1, 1], 8), ([1, 2, 3, 2], 8), ([1, 2, 4, 1], 8), ([1, 2, 5], 8), ([1, 3, 1, 1, 1, 1], 8), ([1, 3, 1, 1, 2], 8), ([1, 3, 1, 2, 1], 8), ([1, 3, 1, 3], 8), ([1, 3, 2, 1, 1], 8), ([1, 3, 2, 2], 8), ([1, 3, 3, 1], 8), ([1, 3, 4], 8), ([1, 4, 1, 1, 1], 8), ([1, 4, 1, 2], 8), ([1, 4, 2, 1], 8), ([1, 4, 3], 8), ([1, 5, 1, 1], 8), ([1, 5, 2], 8), ([1, 6, 1], 8), ([1, 7], 8), ([2, 1, 1, 1, 1, 1, 1], 8), ([2, 1, 1, 1, 1, 2], 8), ([2, 1, 1, 1, 2, 1], 8), ([2, 1, 1, 1, 3], 8), ([2, 1, 1, 2, 1, 1], 8), ([2, 1, 1, 2, 2], 8), ([2, 1, 1, 3, 1], 8), ([2, 1, 1, 4], 8), ([2, 1, 2, 1, 1, 1], 8), ([2, 1, 2, 1, 2], 8), ([2, 1, 2, 2, 1], 8), ([2, 1, 2, 3], 8), ([2, 1, 3, 1, 1], 8), ([2, 1, 3, 2], 8), ([2, 1, 4, 1], 8), ([2, 1, 5], 8), ([2, 2, 1, 1, 1, 1], 8), ([2, 2, 1, 1, 2], 8), ([2, 2, 1, 2, 1], 8), ([2, 2, 1, 3], 8), ([2, 2, 2, 1, 1], 8), ([2, 2, 2, 2], 8), ([2, 2, 3, 1], 8), ([2, 2, 4], 8), ([2, 3, 1, 1, 1], 8), ([2, 3, 1, 2], 8), ([2, 3, 2, 1], 8), ([2, 3, 3], 8), ([2, 4, 1, 1], 8), ([2, 4, 2], 8), ([2, 5, 1], 8), ([2, 6], 8), ([3, 1, 1, 1, 1, 1], 8), ([3, 1, 1, 1, 2], 8), ([3, 1, 1, 2, 1], 8), ([3, 1, 1, 3], 8), ([3, 1, 2, 1, 1], 8), ([3, 1, 2, 2], 8), ([3, 1, 3, 1], 8), ([3, 1, 4], 8), ([3, 2, 1, 1, 1], 8), ([3, 2, 1, 2], 8), ([3, 2, 2, 1], 8), ([3, 2, 3], 8), ([3, 3, 1, 1], 8), ([3, 3, 2], 8), ([3, 4, 1], 8), ([3, 5], 8), ([4, 1, 1, 1, 1], 8), ([4, 1, 1, 2], 8), ([4, 1, 2, 1], 8), ([4, 1, 3], 8), ([4, 2, 1, 1], 8), ([4, 2, 2], 8), ([4, 3, 1], 8), ([4, 4], 8), ([5, 1, 1, 1], 8), ([5, 1, 2], 8), ([5, 2, 1], 8), ([5, 3], 8), ([6, 1, 1], 8), ([6, 2], 8), ([7, 1], 8), ([8], 8)]def permutations_N_coins_with_sum_M(N,M, logging=False):
possibles = []
depth_limit = N
def multi_loop(K, vals, depth):
if K < 1:
if depth == depth_limit:
possibles.append((vals, sum(vals)))
return
if depth == depth_limit:
return
for i in range(1,K+1):
multi_loop(K-i, vals + [i], depth+1)
multi_loop(K=M, vals=[], depth=0)
print("- number of permutations about N({}) items with sum M({}) : {}".format(N,M,len(possibles)))
if logging:
print(possibles)Let's run this code...
permutations_N_coins_with_sum_M(N=6, M=0)
permutations_N_coins_with_sum_M(N=6, M=6, logging=True)
permutations_N_coins_with_sum_M(N=6, M=10, logging=True)
permutations_N_coins_with_sum_M(N=6, M=20, logging=False)Output is ...
- number of permutations about N(6) items with sum M(0) : 0
- number of permutations about N(6) items with sum M(6) : 1
[([1, 1, 1, 1, 1, 1], 6)]
- number of permutations about N(6) items with sum M(10) : 126
[([1, 1, 1, 1, 1, 5], 10), ([1, 1, 1, 1, 2, 4], 10), ([1, 1, 1, 1, 3, 3], 10), ([1, 1, 1, 1, 4, 2], 10), ([1, 1, 1, 1, 5, 1], 10), ([1, 1, 1, 2, 1, 4], 10), ([1, 1, 1, 2, 2, 3], 10), ([1, 1, 1, 2, 3, 2], 10), ([1, 1, 1, 2, 4, 1], 10), ([1, 1, 1, 3, 1, 3], 10), ([1, 1, 1, 3, 2, 2], 10), ([1, 1, 1, 3, 3, 1], 10), ([1, 1, 1, 4, 1, 2], 10), ([1, 1, 1, 4, 2, 1], 10), ([1, 1, 1, 5, 1, 1], 10), ([1, 1, 2, 1, 1, 4], 10), ([1, 1, 2, 1, 2, 3], 10), ([1, 1, 2, 1, 3, 2], 10), ([1, 1, 2, 1, 4, 1], 10), ([1, 1, 2, 2, 1, 3], 10), ([1, 1, 2, 2, 2, 2], 10), ([1, 1, 2, 2, 3, 1], 10), ([1, 1, 2, 3, 1, 2], 10), ([1, 1, 2, 3, 2, 1], 10), ([1, 1, 2, 4, 1, 1], 10), ([1, 1, 3, 1, 1, 3], 10), ([1, 1, 3, 1, 2, 2], 10), ([1, 1, 3, 1, 3, 1], 10), ([1, 1, 3, 2, 1, 2], 10), ([1, 1, 3, 2, 2, 1], 10), ([1, 1, 3, 3, 1, 1], 10), ([1, 1, 4, 1, 1, 2], 10), ([1, 1, 4, 1, 2, 1], 10), ([1, 1, 4, 2, 1, 1], 10), ([1, 1, 5, 1, 1, 1], 10), ([1, 2, 1, 1, 1, 4], 10), ([1, 2, 1, 1, 2, 3], 10), ([1, 2, 1, 1, 3, 2], 10), ([1, 2, 1, 1, 4, 1], 10), ([1, 2, 1, 2, 1, 3], 10), ([1, 2, 1, 2, 2, 2], 10), ([1, 2, 1, 2, 3, 1], 10), ([1, 2, 1, 3, 1, 2], 10), ([1, 2, 1, 3, 2, 1], 10), ([1, 2, 1, 4, 1, 1], 10), ([1, 2, 2, 1, 1, 3], 10), ([1, 2, 2, 1, 2, 2], 10), ([1, 2, 2, 1, 3, 1], 10), ([1, 2, 2, 2, 1, 2], 10), ([1, 2, 2, 2, 2, 1], 10), ([1, 2, 2, 3, 1, 1], 10), ([1, 2, 3, 1, 1, 2], 10), ([1, 2, 3, 1, 2, 1], 10), ([1, 2, 3, 2, 1, 1], 10), ([1, 2, 4, 1, 1, 1], 10), ([1, 3, 1, 1, 1, 3], 10), ([1, 3, 1, 1, 2, 2], 10), ([1, 3, 1, 1, 3, 1], 10), ([1, 3, 1, 2, 1, 2], 10), ([1, 3, 1, 2, 2, 1], 10), ([1, 3, 1, 3, 1, 1], 10), ([1, 3, 2, 1, 1, 2], 10), ([1, 3, 2, 1, 2, 1], 10), ([1, 3, 2, 2, 1, 1], 10), ([1, 3, 3, 1, 1, 1], 10), ([1, 4, 1, 1, 1, 2], 10), ([1, 4, 1, 1, 2, 1], 10), ([1, 4, 1, 2, 1, 1], 10), ([1, 4, 2, 1, 1, 1], 10), ([1, 5, 1, 1, 1, 1], 10), ([2, 1, 1, 1, 1, 4], 10), ([2, 1, 1, 1, 2, 3], 10), ([2, 1, 1, 1, 3, 2], 10), ([2, 1, 1, 1, 4, 1], 10), ([2, 1, 1, 2, 1, 3], 10), ([2, 1, 1, 2, 2, 2], 10), ([2, 1, 1, 2, 3, 1], 10), ([2, 1, 1, 3, 1, 2], 10), ([2, 1, 1, 3, 2, 1], 10), ([2, 1, 1, 4, 1, 1], 10), ([2, 1, 2, 1, 1, 3], 10), ([2, 1, 2, 1, 2, 2], 10), ([2, 1, 2, 1, 3, 1], 10), ([2, 1, 2, 2, 1, 2], 10), ([2, 1, 2, 2, 2, 1], 10), ([2, 1, 2, 3, 1, 1], 10), ([2, 1, 3, 1, 1, 2], 10), ([2, 1, 3, 1, 2, 1], 10), ([2, 1, 3, 2, 1, 1], 10), ([2, 1, 4, 1, 1, 1], 10), ([2, 2, 1, 1, 1, 3], 10), ([2, 2, 1, 1, 2, 2], 10), ([2, 2, 1, 1, 3, 1], 10), ([2, 2, 1, 2, 1, 2], 10), ([2, 2, 1, 2, 2, 1], 10), ([2, 2, 1, 3, 1, 1], 10), ([2, 2, 2, 1, 1, 2], 10), ([2, 2, 2, 1, 2, 1], 10), ([2, 2, 2, 2, 1, 1], 10), ([2, 2, 3, 1, 1, 1], 10), ([2, 3, 1, 1, 1, 2], 10), ([2, 3, 1, 1, 2, 1], 10), ([2, 3, 1, 2, 1, 1], 10), ([2, 3, 2, 1, 1, 1], 10), ([2, 4, 1, 1, 1, 1], 10), ([3, 1, 1, 1, 1, 3], 10), ([3, 1, 1, 1, 2, 2], 10), ([3, 1, 1, 1, 3, 1], 10), ([3, 1, 1, 2, 1, 2], 10), ([3, 1, 1, 2, 2, 1], 10), ([3, 1, 1, 3, 1, 1], 10), ([3, 1, 2, 1, 1, 2], 10), ([3, 1, 2, 1, 2, 1], 10), ([3, 1, 2, 2, 1, 1], 10), ([3, 1, 3, 1, 1, 1], 10), ([3, 2, 1, 1, 1, 2], 10), ([3, 2, 1, 1, 2, 1], 10), ([3, 2, 1, 2, 1, 1], 10), ([3, 2, 2, 1, 1, 1], 10), ([3, 3, 1, 1, 1, 1], 10), ([4, 1, 1, 1, 1, 2], 10), ([4, 1, 1, 1, 2, 1], 10), ([4, 1, 1, 2, 1, 1], 10), ([4, 1, 2, 1, 1, 1], 10), ([4, 2, 1, 1, 1, 1], 10), ([5, 1, 1, 1, 1, 1], 10)]
- number of permutations about N(6) items with sum M(20) : 11628