III. If 7 cards are dealt from an ordinary deck of 52 playing cards, what is the probability that e
be face cards?
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Jan 23,2025

Question

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

UpStudy AI · Accepted Answer

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 $$\frac{99}{16723070}$$.
The probability of drawing 7 face cards from a deck of 52 playing cards is approximately $$5.92 \times 10^{-6}$$ or $$\frac{99}{16723070}$$.

III. If 7 cards are dealt from an ordinary deck of 52 playing cards, what is the probability that e
be face cards?
Ans.

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