Answer :
Answer: d) 6,534,385
Therefore, the number of ways of selecting 9 hearts is 6,534,385
Step-by-step explanation:
Given;
Number of cards to be chosen = 12
Number of hearts to be chosen = 9
Number of non-hearts to be selected = 3
Total number of cards = 52
Number of hearts total = 13
Number of non-hearts total= 39
The number of ways of selecting hearts can be given by the illustration below.
Number of ways of selecting hearts × number of ways of selecting non-hearts
N = 13C9 × 39C3 (it's combination since order is not important)
N = 13!/(9! ×4!) × 39!/(36! × 3!)
N = 6,534,385
Therefore, the number of ways of selecting 9 hearts is 6,534,385
The number of ways to choose 9 hearts from 12 cards is 220, option (b) is correct.
To solve this problem, we'll use the combination formula, which is used when selecting objects from a larger set without regard to the order. The formula for combinations is:
[tex]\[ \text{C}(n, k) = \frac{n!}{k!(n - k)!} \][/tex]
where [tex]\( n \)[/tex] is the total number of items, [tex]\( k \)[/tex] is the number of items to choose, and (!) denotes factorial.
Given that we need to choose 12 cards from a deck of 52, and we want to know the number of ways to choose 9 hearts, which is [tex]\( k = 9 \)[/tex] hearts out of a total of [tex]\( n = 12 \)[/tex] cards.
Step 1:
Calculate the number of ways to choose 9 hearts from the 12 chosen cards:
[tex]\[ \text{C}(12, 9) = \frac{12!}{9!(12 - 9)!} \][/tex]
Step 2:
Simplify the expression:
[tex]\[ \text{C}(12, 9) = \frac{12!}{9!3!} \][/tex]
[tex]\[ \text{C}(12, 9) = \frac{12 \times 11 \times 10}{3 \times 2 \times 1} \][/tex]
[tex]\[ \text{C}(12, 9) = \frac{1320}{6} \][/tex]
[tex]\[ \text{C}(12, 9) = 220 \][/tex]
Therefore, there are 220 ways to choose 9 hearts from 12 cards. So, the correct answer is option (b).
