III. If 7 cards are dealt from an ordinary deck of 52 playing cards, what is the probability that e be face cards? Ans. \( \quad n=\binom{52}{7}=133784560 \)

To find the probability of drawing 7 face cards from a deck of 52 playing cards, we first need to determine the total number of ways to draw 7 cards from the deck and then find the number of ways to draw 7 face cards. Given: - Total number of cards in the deck = 52 - Number of face cards in the deck = 12 (since there are 4 suits with 3 face cards each) 1. Calculate the total number of ways to draw 7 cards from the deck: \[ \text{Total ways} = \binom{52}{7} \] 2. Calculate the number of ways to draw 7 face cards from the deck: \[ \text{Ways to draw 7 face cards} = \binom{12}{7} \] 3. Calculate the probability of drawing 7 face cards: \[ \text{Probability} = \frac{\text{Ways to draw 7 face cards}}{\text{Total ways}} \] Let's calculate the values: Calculate the value by following steps: - step0: Calculate: \( { }_{52}C_{7}\) - step1: Expand the expression: \(\frac{52!}{7!\times \left(52-7\right)!}\) - step2: Subtract the numbers: \(\frac{52!}{7!\times 45!}\) - step3: Calculate: \(\frac{52!}{5040\times 45!}\) - step4: Factor the expression: \(\frac{52\times 51\times 50\times 49\times 48\times 47\times 46\times 45!}{5040\times 45!}\) - step5: Reduce the fraction: \(13\times 17\times 5\times 7\times 8\times 47\times 46\) - step6: Multiply the terms: \(221\times 5\times 7\times 8\times 47\times 46\) - step7: Multiply the terms: \(1105\times 7\times 8\times 47\times 46\) - step8: Multiply the terms: \(7735\times 8\times 47\times 46\) - step9: Multiply the terms: \(61880\times 47\times 46\) - step10: Multiply the terms: \(2908360\times 46\) - step11: Multiply the numbers: \(133784560\) Calculate or simplify the expression \( \binom{12}{7} \). Calculate the value by following steps: - step0: Calculate: \( { }_{12}C_{7}\) - step1: Expand the expression: \(\frac{12!}{7!\times \left(12-7\right)!}\) - step2: Subtract the numbers: \(\frac{12!}{7!\times 5!}\) - step3: Expand the expression: \(\frac{12\times 11\times 10\times 9\times 8\times 7!}{7!\times 5!}\) - step4: Simplify: \(\frac{12\times 11\times 10\times 9\times 8}{5!}\) - step5: Multiply: \(\frac{95040}{5!}\) - step6: Calculate the value: \(\frac{95040}{120}\) - step7: Reduce the fraction: \(792\) Calculate or simplify the expression \( \frac{792}{133784560} \). Calculate the value by following steps: - step0: Calculate: \(\frac{792}{133784560}\) - step1: Reduce the fraction: \(\frac{99}{16723070}\) The probability of drawing 7 face cards from a deck of 52 playing cards is approximately \(5.919966 \times 10^{-6}\) or \(\frac{99}{16723070}\).