Q. in how many ways can 7 identical balls be distributed in 5 different boxes if any box can contain any number of balls?
Answers
Answer: 330 different ways
Step-by-step explanation:
I'm assuming you already know the basic of permutation and combination.
Case 1: Only one of the boxes are filled (0 0 0 0 7). In this case there are 5 different ways or 5C1 * 5.
Case 2: Two of the boxes are filled with three different combinations (0 0 0 1 6, 0 0 0 2 5, 0 0 0 3 4). Ignore the zeros and check how many possible ways for each combination (e.g. 0 0 0 1 6 will get you 1 6 and 6 1). You'll get 5C2 * 6 different ways for this case.
Case 3: Three of the boxes (1 1 5 [3 possible ways], 1 2 4 [6], 2 2 3 [3], 1 3 3 [3]), 5C3 * 15.
Do the same for case 4-5. You'll get the following expression:
5C1 * 5 + 5C2 * 6 + 5C3 * 15 + 5C4 * 20 + 5C5 * 15
= 1*5 + 10*6 + 10*15 + 5*20 + 1*15 = 330 different ways
Hope this helps.