Question

The number of 5-card combinations out of a deck of 52 cards if there is exactly one ace in each combination is:

• A. 823764
• B. 363428
• C. 778320
• D. 542380

Hint:

For combinations with a specific element, first choose that element and then choose the remaining from the rest.

Answer 1

The Correct Option is C

Step 1: Split the hand into the ace and the rest.
A valid hand needs exactly one of the 4 aces, plus the remaining 4 cards drawn from the 48 non-ace cards.

Step 2: Count each part separately.
Choosing the ace: \(\binom{4}{1}=4\)
Choosing the other four cards: \(\binom{48}{4}=\frac{48\times47\times46\times45}{4\times3\times2\times1}=194580\)

Step 3: Multiply the independent choices.
\[ 4\times194580=778320 \]

Step 4: Conclusion.
\[ \boxed{778320} \]