Question

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Answer 1

Using the combination formula, the probabilities are given as follows:

a) P(4 seniors) = 0.0008.

b) P(1 of each) = 0.0999.

c) P(2 sophomore, 2 freshmen) = 0.0225.

d) P (at least one senior) = 0.5866.

Combination Formula

$C_{n,x}$ is the number of different combinations of x objects from a set of n elements, given by the formula presented below, involving factorials.

$C_{n,x} = \frac{n!}{x!(n-x)!}$

In this problem, 4 captains are taken from a set of 13 + 13 + 10 + 16 = 52, hence the number of outcomes is:

$C_{52,4} = \frac{52!}{4!48!} = 270725$

The number of outcomes with four seniors is given as follows:

$C_{10,4} = \frac{10!}{4!6!} = 210$

Hence the probability is:

p = 210/270725 = 0.0008.

With no seniors, the number of outcomes is:

$C_{42,4} = \frac{42!}{4!38!} = 111930$

Hence with at least one senior, the number of outcomes is:

270725 - 111930 = 158795.

Hence the probability is:

p = 158795/270725 = 0.5866.

With one of each, the number of outcomes is:

13 x 13 x 10 x 16 = 27040

Then the probability is:

27040/270725 = 0.0999.

With two sophomores and two freshmen, the number of outcomes is:

$C_{13,2} \times C_{13,2} = \frac{13!}{2!11!} \times \frac{13!}{2!11!} = 6084$

Then the probability is:

6084/270725 = 0.0225.

More can be learned about the combination formula at brainly.com/question/25821700

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