Question

1. a 7 - card hand is dealt from a 52 - card deck. which expression represents the number of possible hands? a. ( 52cdot7 ) b. ( 52! ) c. ( 52p7 ) d. ( 52c7 ) 4. how many distinct arrangements of the letters in probability (11 letters, with repeats) are possible? a. ( 1,663,200 ) b. ( 7,241,600 ) c. ( 9,979,200 ) d. ( 39,916,800 )

Explanation:

Question 3

Step1: Understand combination and permutation

Combination \(C(n,r)=\frac{n!}{r!(n - r)!}\) (order doesn't matter). Permutation \(P(n,r)=\frac{n!}{(n - r)!}\) (order matters). Here, a 7 - card hand (order of cards in hand doesn't matter) is a combination problem. \(n = 52\) (total cards), \(r=7\) (cards in hand).

Step2: Analyze each option

• Option A: \(52\times7\) is wrong as it's not a standard counting formula for hands.
• Option B: \(52!\) is the factorial of 52 (number of ways to arrange 52 distinct objects), not for 7 - card hands.
• Option C: \(_{52}P_7=\frac{52!}{(52 - 7)!}\) (order matters, not for hands).
• Option D: \(_{52}C_7=\frac{52!}{7!(52 - 7)!}\) (order doesn't matter, correct for 7 - card hands).

Step1: Use the formula for permutations of multi - set

The formula for the number of permutations of a multi - set with \(n\) objects, where there are \(n_1\) of one kind, \(n_2\) of another kind, \(\cdots\), \(n_k\) of the \(k\)th kind is \(\frac{n!}{n_1!n_2!\cdots n_k!}\). For the word "PROBABILITY", \(n = 11\) letters. The letter 'B' appears \(2\) times, the letter 'I' appears \(2\) times.

Step2: Calculate the number of permutations

$$
LATEXBLOCK0
$$

Answer:

D. \(52C7\)

Question 4