1. In how many ways can we place r red balls and w white balls in n boxes so tha

Question

1. In how many ways can we place r red balls and w white balls in n boxes so that each box conta ins one ball of each color? 2. (a) How many sequences (lists) of m Os and n 1s are there b) How many sequences are there in which each 1 is separated by at least two 0s? (Assume that for this part m 2 2(n - 1).) 3. We are given a red box, a blue box and a green box. We are also given 10 red balls, 10 blue balls, and 10 green balls. Balls of the same colour are indistinguishable. Consider the following constraints: A:No box contains a ball that has the same colour as the box B: No box is empty. Determine the number of ways in which we can put 30 balls into boxes so that: (a) No constraint has to be satisfied. Every combination is allowed (b) Constraint A is satisfied (c) Constraint B is satisfied (d) Constraints A and B are satisfied.

Explanation / Answer

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Q1) Place r red balls and w white balls in n boxes so that each box contains one ball of each color.

Answer)

Assuming that the boxes are distinct and the balls are identical i.e. all ed balls are same, all white balls are same.

r = number of red balls

w = number of white balls.

Assuming, At most one in each box,

Place r red balls in n boxes so that each box contains one ball of red =

n C r

Place w white balls in n boxes so that each box contains one ball of white =

n C w

Thus the number of ways to Place r red balls and w white balls in n boxes so that each box contains one ball of each color =

(n C r) X (n C w)

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