In how many ways can a hand consisting of 3 spades, 3 hearts, 2 clubs, and 1 diamond be selected from a deck of 52 cards?
A) 38,332,800
B) 168,799,904
C) 82,941,144
D) 1,900,734,550
E) 3,770,052

Question

In how many ways can a hand consisting of 3 spades, 3 hearts, 2 clubs, and 1 diamond be selected from a deck of 52 cards? ​
A) 38,332,800
B) 168,799,904
C) 82,941,144
D) 1,900,734,550
E) 3,770,052

Quizplus · Accepted Answer

To find the number of ways to select the hand, we can use the formula for combinations.

Number of ways to select 3 spades: $\binom{13}{3}$ 
Number of ways to select 3 hearts: $\binom{13}{3}$ 
Number of ways to select 2 clubs: $\binom{13}{2}$ 
Number of ways to select 1 diamond: $\binom{13}{1}$

By the Multiplication Principle, the total number of ways to select the hand is the product of these values:

$$\binom{13}{3} \cdot \binom{13}{3} \cdot \binom{13}{2} \cdot \binom{13}{1} = 82,941,144$$

Therefore, the correct answer is C.

In How Many Ways Can a Hand Consisting of 3