The U.S. Senate has two senators from each of the 50 states (100 senators total). A)...

The U.S. Senate has two senators from each of the 50 states (100 senators total).

A) In how many ways can a committee of five senators be chosen if no state is to have two members on the committee?

B) In how many ways can a committee of seven senators be chosen if exactly one pair of senators share their state?

C) What is the probability a randomly selected pair of senators will be from different states?

Solutions

Expert Solution

Divide the 100 senators in two groups such that each group will have only one person representing his state.

Suppose we select 5 from a group this will happen in 50C_5. since we have two groups The total number of ways is 2*50C₅

Suppose we selected one person randomly from a group. This can be done in 50 ways. And the remaining 4 persons from the other group among 49 persons of the second group because a second person from the same state should not be selected. so This will happen in 50*49C₄ ways

Suppose two persons are selected from the first group then this will happen in 50 C₂ ways and remaining 3 need to be selected from the members of remaining 48 states. This will happen in 50 C₂ *48 C₃

Continuing in this manner the total number of ways of selecting 5 senators is

B)After choosing 5 senators in the above manner there will be 45 pairs (states)of senators which donot belong to the committee. Among them any pair can be a paired member in the committee.

The number of ways of selecting 5 senetors*45 wil l be the answer for B)

C) There are 100C₂ ways of selecting a pair from the 100 members senates. Of these 100C₂ ways there will be exactly 50 pairs representing the same state. So the probability that a randomly selected pair of senator will be from different states =

orchestra answered 1 year ago

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