# Question: From a set of n people a committee of size

From a set of n people, a committee of size j is to be chosen, and from this committee, a subcommittee of size i, i ≤ j, is also to be chosen.

(a) Derive a combinatorial identity by computing, in two ways, the number of possible choices of the committee and subcommittee—first by supposing that the committee is chosen first and then the subcommittee is chosen, and second by supposing that the subcommittee is chosen first and then the remaining members of the committee are chosen.

(b) Use part (a) to prove the following combinatorial identity:

(c) Use part (a) and Theoretical Exercise 13 to show that

(a) Derive a combinatorial identity by computing, in two ways, the number of possible choices of the committee and subcommittee—first by supposing that the committee is chosen first and then the subcommittee is chosen, and second by supposing that the subcommittee is chosen first and then the remaining members of the committee are chosen.

(b) Use part (a) to prove the following combinatorial identity:

(c) Use part (a) and Theoretical Exercise 13 to show that

**View Solution:**## Answer to relevant Questions

Let Hk(n) be the number of vectors x1, . . ., xk for which each xi is a positive integer satisfying 1 ≤ xi ≤ n and x1 ≤ x2 ≤ . . . ≤ xk. (a) Without any computations, argue that H1(n) = n How many vectors are ...In how many ways can n identical balls be distributed into r urns so that the ith urn contains at least mi balls, for each i = 1, . . ., r? Assume that A box contains 3 marbles: 1 red, 1 green, and 1 blue. Consider an experiment that consists of taking 1 marble from the box and then replacing it in the box and drawing a second marble from the box. Describe the sample space. ...In an experiment, die is rolled continually until a 6 appears, at which point the experiment stops. What is the sample space of this experiment? Let En denote the event that n rolls are necessary to complete the experiment. ...Seven balls are randomly withdrawn from an urn that contains 12 red, 16 blue, and 18 green balls. Find the probability that (a) 3 red, 2 blue, and 2 green balls are withdrawn; (b) At least 2 red balls are withdrawn; (c) All ...Post your question