select the correct answer. if 6 cards are drawn at random from a standard deck of 52 cards, what is the…

Explanation:

Step1: Calculate number of ways to choose 2 spades

There are 13 spades in a deck. The number of ways to choose 2 spades out of 13 is given by the combination formula $C(n,k)=\frac{n!}{k!(n - k)!}$, where $n = 13$ and $k=2$. So $C(13,2)=\frac{13!}{2!(13 - 2)!}=\frac{13\times12}{2\times1}=78$.

Step2: Calculate number of ways to choose non - spades

There are $52-13 = 39$ non - spades. We need to choose $6 - 2=4$ non - spades. Using the combination formula with $n = 39$ and $k = 4$, we have $C(39,4)=\frac{39!}{4!(39 - 4)!}=\frac{39\times38\times37\times36}{4\times3\times2\times1}=82251$.

Step3: Calculate number of ways to choose 6 cards from 52

The number of ways to choose 6 cards from 52 is $C(52,6)=\frac{52!}{6!(52 - 6)!}=\frac{52\times51\times50\times49\times48\times47}{6\times5\times4\times3\times2\times1}=20358520$.

Step4: Calculate the probability

The probability $P$ that exactly 2 of the 6 cards are spades is the product of the number of ways to choose 2 spades and the number of ways to choose 4 non - spades divided by the number of ways to choose 6 cards from 52. So $P=\frac{C(13,2)\times C(39,4)}{C(52,6)}=\frac{78\times82251}{20358520}\approx0.315$.

Answer:

C. 0.315