there are $n \ge 4$ people in a city. And the city has its officials, consisting of 1 mayor and 3 vice-mayors. The entire board consists of 4 distinct students. Prove that by counting. In 2 different methods, count the number of ways to choose a board of officials

$$n \binom{n-1}{3} = (n-3)\binom{n}{3}$$

Any tips on two ways to help me started? I am stuck on this question and just need some starting help!

  • $\begingroup$ Can you go to the definition $\binom{n}{k}$ = $\frac{n!}{k!(n-k)!}$? $\endgroup$ – graydad Sep 28 '14 at 17:22
  • $\begingroup$ Rolled back destructive edit. $\endgroup$ – apnorton Oct 1 '14 at 0:03

Think of this: you can either choose the mayor first, THEN choose the vice-mayors, OR choose the vice-mayors first and THEN choose the mayor. Do you see how these give the two expressions you gave above?


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