Four cards are drawn at random from an ordinary deck of 52 cards. Find the probability that the 4 -card hand contains no more than 1 spade. How many ways can a hand with no more than 1 spade be drawn? ways (Type a whole number.) The probability that the 4-card hand contains no more than 1 spade is (Round to four decimal places as needed.)

To find the number of ways to draw a 4-card hand with no more than 1 spade from a standard deck of 52 cards, we need to consider two cases: the hands with 0 spades and hands with 1 spade. 1. **Case 1: 0 spades** In this case, all 4 cards must come from the 39 non-spade cards. The number of ways to choose 4 cards from these 39 is given by the combination formula \( \binom{n}{r} \): \[ \binom{39}{4} = \frac{39 \times 38 \times 37 \times 36}{4 \times 3 \times 2 \times 1} = 82251 \] 2. **Case 2: 1 spade** We want to choose 1 spade from the 13 available spades and 3 cards from the 39 non-spade cards. The number of ways to choose 1 spade is \( \binom{13}{1} \) and for 3 non-spades is \( \binom{39}{3} \): \[ \binom{13}{1} = 13 \] \[ \binom{39}{3} = \frac{39 \times 38 \times 37}{3 \times 2 \times 1} = 9139 \] Thus, the total for this case is: \[ 13 \times 9139 = 118807 \] Now, we can compute the total number of ways to draw a 4-card hand with no more than 1 spade: \[ 82251 + 118807 = 201058 \] Next, we calculate the total number of 4-card hands that can be drawn from a 52-card deck, which is: \[ \binom{52}{4} = \frac{52 \times 51 \times 50 \times 49}{4 \times 3 \times 2 \times 1} = 270725 \] Finally, the probability that the 4-card hand contains no more than 1 spade is the number of favorable hands divided by the total number of hands: \[ \text{Probability} = \frac{201058}{270725} ≈ 0.7422 \] So, the answers are as follows: The number of ways a hand with no more than 1 spade can be drawn is: **201058** The probability that the 4-card hand contains no more than 1 spade is: **0.7422** (rounded to four decimal places).