Committee selection with no two consecutive people. 1. Assume that $10$ people are sitting around a round table. Determine the number of ways to choose a committee, where the committee is made up of two people who are NOT sitting next to each other. 2. Assume that $10$ people are sitting around a round table. Determine the number of ways to choose a committee, where the committee is made up of three people, Person X and two other people, such that NO two of the three people are sitting next to each other.

1. Possible committees - Committees disallowed = $^{10}C_2$ - 10

2. Assuming Person X is fixed, we can have to pick 2 members from the 7 that are neither X nor sitting beside X. This gives $^7C_2 - 6$ since we have to discount the committees made up of neighbours