Question

from a standard 52-card deck, how many 9-card hands consist of 5 spades and 4 diamonds? there are \\(\square\\) possible 9-card hands consisting of 5 spades and 4 diamonds. (type a whole number.)

Explanation:

Step1: Calculate ways to choose 5 spades

A standard deck has 13 spades. The number of ways to choose 5 spades is given by the combination formula $C(n,k)=\frac{n!}{k!(n-k)!}$, so $C(13,5)$.
$C(13,5)=\frac{13!}{5!8!}=\frac{13×12×11×10×9}{5×4×3×2×1}=1287$

Step2: Calculate ways to choose 4 diamonds

A standard deck has 13 diamonds. The number of ways to choose 4 diamonds is $C(13,4)$.
$C(13,4)=\frac{13!}{4!9!}=\frac{13×12×11×10}{4×3×2×1}=715$

Step3: Multiply the two results

Total hands = ways to choose spades × ways to choose diamonds = $1287×715$.
$1287×715 = 920,205$

Answer:

920205