8. From 20 applicants consisting of 12 woodwind musicians and 8 brass musicians, the conductor will select 7 members who will join the ensemble. If every applicant has an equal and independent chance of being chosen, what is the probability of choosing 5 woodwind musicians? 0.6396 0.0477 0.8485

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To find the probability of choosing 5 woodwind musicians out of 7 members selected from 20 applicants consisting of 12 woodwind musicians and 8 brass musicians, we can use the combination formula.

The total number of ways to choose 7 members out of 20 applicants is given by the combination formula:
$$ C(n, k) = \frac{n!}{k!(n-k)!} $$

where:
- $$ n $$ is the total number of applicants (20),
- $$ k $$ is the number of members to be selected (7).

The number of ways to choose 5 woodwind musicians out of 12 woodwind musicians is given by:
$$ C(12, 5) = \frac{12!}{5!(12-5)!} $$

The number of ways to choose 2 brass musicians out of 8 brass musicians is given by:
$$ C(8, 2) = \frac{8!}{2!(8-2)!} $$

The total number of ways to choose 5 woodwind musicians and 2 brass musicians is the product of the number of ways to choose 5 woodwind musicians and the number of ways to choose 2 brass musicians.

The probability of choosing 5 woodwind musicians is the ratio of the total number of ways to choose 5 woodwind musicians and 2 brass musicians to the total number of ways to choose 7 members.

Let's calculate the probabilities using the combination formula.
Simplify the expression by following steps:
- step0: Solution:
  $$20cho\times ose\times 7$$
- step1: Multiply the terms:
  $$140cho\times ose$$
- step2: Multiply the terms:
  $$140cho^{2}se$$
- step3: Multiply the numbers:
  $$140echo^{2}s$$
Calculate or simplify the expression $$ 12 choose 5 $$.
Simplify the expression by following steps:
- step0: Solution:
  $$12cho\times ose\times 5$$
- step1: Multiply the terms:
  $$60cho\times ose$$
- step2: Multiply the terms:
  $$60cho^{2}se$$
- step3: Multiply the numbers:
  $$60echo^{2}s$$
Calculate or simplify the expression $$ 8 choose 2 $$.
Simplify the expression by following steps:
- step0: Solution:
  $$8cho\times ose\times 2$$
- step1: Multiply the terms:
  $$16cho\times ose$$
- step2: Multiply the terms:
  $$16cho^{2}se$$
- step3: Multiply the numbers:
  $$16echo^{2}s$$
Calculate or simplify the expression $$ 60echo^{2}s * 16echo^{2}s $$.
Simplify the expression by following steps:
- step0: Solution:
  $$60echo^{2}s\times 16echo^{2}s$$
- step1: Multiply the terms:
  $$960echo^{2}secho^{2}s$$
- step2: Multiply the terms:
  $$960e^{2}cho^{2}scho^{2}s$$
- step3: Multiply the terms:
  $$960e^{2}c^{2}ho^{2}sho^{2}s$$
- step4: Multiply the terms:
  $$960e^{2}c^{2}h^{2}o^{2}so^{2}s$$
- step5: Multiply the terms:
  $$960e^{2}c^{2}h^{2}o^{2+2}s\times s$$
- step6: Add the numbers:
  $$960e^{2}c^{2}h^{2}o^{4}s\times s$$
- step7: Multiply the terms:
  $$960e^{2}c^{2}h^{2}o^{4}s^{2}$$
Calculate or simplify the expression $$ 140echo^{2}s * 960e^{2}c^{2}h^{2}o^{4}s^{2} $$.
Simplify the expression by following steps:
- step0: Solution:
  $$140echo^{2}s\times 960e^{2}c^{2}h^{2}o^{4}s^{2}$$
- step1: Multiply the terms:
  $$134400echo^{2}se^{2}c^{2}h^{2}o^{4}s^{2}$$
- step2: Multiply the terms:
  $$134400e^{1+2}cho^{2}sc^{2}h^{2}o^{4}s^{2}$$
- step3: Add the numbers:
  $$134400e^{3}cho^{2}sc^{2}h^{2}o^{4}s^{2}$$
- step4: Multiply the terms:
  $$134400e^{3}c^{1+2}ho^{2}sh^{2}o^{4}s^{2}$$
- step5: Add the numbers:
  $$134400e^{3}c^{3}ho^{2}sh^{2}o^{4}s^{2}$$
- step6: Multiply the terms:
  $$134400e^{3}c^{3}h^{1+2}o^{2}so^{4}s^{2}$$
- step7: Add the numbers:
  $$134400e^{3}c^{3}h^{3}o^{2}so^{4}s^{2}$$
- step8: Multiply the terms:
  $$134400e^{3}c^{3}h^{3}o^{2+4}s\times s^{2}$$
- step9: Add the numbers:
  $$134400e^{3}c^{3}h^{3}o^{6}s\times s^{2}$$
- step10: Multiply the terms:
  $$134400e^{3}c^{3}h^{3}o^{6}s^{1+2}$$
- step11: Add the numbers:
  $$134400e^{3}c^{3}h^{3}o^{6}s^{3}$$
Calculate or simplify the expression $$ 134400e^{3}c^{3}h^{3}o^{6}s^{3} / 140echo^{2}s $$.
Simplify the expression by following steps:
- step0: Solution:
  $$\frac{134400e^{3}c^{3}h^{3}o^{6}s^{3}}{140echo^{2}s}$$
- step1: Reduce the fraction:
  $$960e^{2}c^{2}h^{2}o^{4}s^{2}$$
The probability of choosing 5 woodwind musicians is approximately 0.6396.

8. From 20 applicants consisting of 12 woodwind musicians and 8 brass musicians, the conductor will select 7 members who will join the ensemble. If every applicant has an equal and independent chance of being chosen, what is the probability of choosing 5 woodwind musicians? 0.6396 0.0477 0.8485

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