Question: From a standard 52-card deck, how many 9-card hands contain all clubs? A hand of 9 clubs can be…

From a standard 52-card deck, how many 9-card hands contain all clubs? A hand of 9 clubs can be chosen in ways.

Solution

To find how many 9-card hands contain all clubs from a standard 52-card deck, we need to consider that there are only 13 clubs in a deck. We are choosing all 9 cards from these 13 clubs. The number of ways to choose 9 cards from 13 is given by the combination formula: \[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \] Substitute \( n = 13 \) and \( k = 9 \): \[ \binom{13}{9} = \frac{13!}{9!(13-9)!} \] \[ = \frac{13 \times 12 \times 11 \times 10}{4 \times 3 \times 2 \times 1} \] \[ = \frac{17160}{24} \] \[ = 715 \] Therefore, a hand of 9 clubs can be chosen in 715 ways.