Question

The number of ways to distribute 30 identical candies among four children C1 , C2 , C3 and C4 so that C2 receives at least 4 and at most 7 candies, C3 receives at least 2 and at most 6 candies, is equal to

(A) 205

(B) 615

(C) 510

(D) 430

Answer 1

To distribute the candies among the four children according to the given conditions, we can use a combination of dividing and using the concept of stars and bars. By selecting the positions for the dividers in the distribution, we can calculate that there are 17550 ways to distribute the candies.

Step-by-step explanation:

To solve this problem, we can use a combination of dividing the candies among the children and using the concept of stars and bars. Let's break it down step-by-step:

1. First, we distribute the minimum number of candies required to C2 and C3 (4 and 2, respectively). This leaves us with 30 - 4 - 2 = 24 candies remaining.
2. Next, we distribute the remaining candies among the four children.

Now, we need to find the number of ways to distribute the candies. We can use the concept of stars and bars here. We have 24 candies and 4 children, which can be represented as:

OO|OOO|OOOO|OOOOO

We need to place the dividers (|) in a way that represents the distribution of candies among the children. Since the candies are identical, the order of the dividers doesn't matter.

Now, we have 24 + 4 - 1 = 27 slots (24 candies + 4 dividers - 1 space). To select the positions for the dividers, we choose 4 slots out of the 27, which can be calculated as:

C(27, 4) = 27! / (4! * (27-4)!)

This simplifies to:

27! / (4! * 23!) = 27 * 26 * 25 * 24 / (4 * 3 * 2 * 1) = 27 * 26 * 25 / (3 * 2 * 1) = 25 * 26 * 27 = 17550

So, the number of ways to distribute the candies is 17550.