Question 385820: Allen is in the Boy Scouts. He has three weeks left in which to complete the hours required for a merit badge in community service. Because of his paper route and homework, he can't put in more than 5 hours in any one week for the merit badge. How many different ways can Allen work a total of 10 hours over the three week period? So far I have 23 different combinations. I am wondering if that is the total or if there is more.
Found 2 solutions by scott8148, sudhanshu_kmr:
Answer by scott8148(6628)   (Show Source):
You can put this solution on YOUR website! you might have too many
each of the 3 weeks must be 5 or less , and they sum to 10
5, 0, 5
5, 1, 4
5, 2, 3
5, 3, 2
5, 4, 1
5, 5, 0
4, 1, 5
4, 2, 4
4, 3, 3
4, 4, 2
4, 5, 1
3, 2, 5
3, 3, 4
3, 4, 3
3, 5, 2
2, 3, 5
2, 4, 4
2, 5, 3
1, 4, 5
1, 5, 4
0, 5, 5
Answer by sudhanshu_kmr(1152)  (Show Source):
You can put this solution on YOUR website! 10 hours can be sum of either of these set (according to question)...
( 5,5,0 ), (5,1,4), (5,2,3), (4,2,4) and (4,3,3)
now,
(5,5,0) can be arrange in 3 weeks by 3!/2! ways = 3
similarly, for(5,1,4)............ ....by 3! ways = 6
......for (5,2,3) ........................ 3! ways = 6
......for (4,2,4).......................3!/2! ways = 3
.....for (4,3,3)........................3!/2! ways = 3
total no. of ways.................................= 21 (sum of above)
so, total 21 ways......
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