At one of George Washington's parties, each man shook hands with everyone except his spouse, and no handshakes took place between women. If \(13\) married couples attended, how many handshakes were there among these \(26\) people? $$\text{(A)} \ 78 \qquad \text{(B)} \ 185 \qquad \text{(C)} \ 234 \qquad \text{(D)} \ 312 \qquad \text{(E)} \ 325$$
Tags: counting,combinatorics,binomial-coefficient
Source: AMC 12 1990 A Q16
Elo Rating: 1500.00
Nine chairs in a row are to be occupied by six students and Professors Alpha, Beta and Gamma. These three professors arrive before the six students and decide to choose their chairs so that each professor will be between two students. In how many ways can Professors Alpha, Beta and Gamma choose their chairs? $$ \textbf{(A)}\ 12 \qquad\textbf{(B)}\ 36 \qquad\textbf{(C)}\ 60 \qquad\textbf{(D)}\ 84 \qquad\textbf{(E)}\ 630 $$
Tags: combinatorics,permutations,cases,counting,casework
Source: AMC 12 1994 A Q22
Elo Rating: 1516.00