five cards are drawn randomly from a standard deck of 52 cards. determine the probability that exactly 3 of…

Explanation:

Step1: Calculate total number of ways to draw 5 - card hands

The number of combinations of choosing (n) items from (m) items is given by the formula (C(m,n)=\frac{m!}{n!(m - n)!}). Here, (m = 52) and (n=5), so the total number of 5 - card hands is (C(52,5)=\frac{52!}{5!(52 - 5)!}=\frac{52\times51\times50\times49\times48}{5\times4\times3\times2\times1}=2598960).

Step2: Calculate number of ways to choose 3 aces

There are 4 aces in a deck. The number of ways to choose 3 aces out of 4 is (C(4,3)=\frac{4!}{3!(4 - 3)!}=\frac{4!}{3!1!}=4).

Step3: Calculate number of ways to choose the remaining 2 non - aces

There are (52-4 = 48) non - aces. The number of ways to choose 2 non - aces out of 48 is (C(48,2)=\frac{48!}{2!(48 - 2)!}=\frac{48\times47}{2\times1}=1128).

Step4: Calculate number of 5 - card hands with exactly 3 aces

By the multiplication principle, the number of 5 - card hands with exactly 3 aces is the product of the number of ways to choose 3 aces and the number of ways to choose 2 non - aces. So, the number of 5 - card hands with exactly 3 aces is (C(4,3)\times C(48,2)=4\times1128 = 4512).

Step5: Calculate the probability

The probability (P) that exactly 3 of the 5 cards are aces is (P=\frac{C(4,3)\times C(48,2)}{C(52,5)}=\frac{4512}{2598960}\approx0.00174).

Answer:

0.00174