Question

suppose we want to choose 5 objects, without replacement, from 15 distinct objects.
(a) how many ways can this be done, if the order of the choices is not taken into consideration?
(b) how many ways can this be done, if the order of the choices is taken into consideration?

Explanation:

Part (a)

Step1: Identify the problem type

This is a combination problem since the order of choices is not considered. The formula for combinations is \( C(n, k)=\frac{n!}{k!(n - k)!} \), where \( n = 15 \) (total objects) and \( k = 5 \) (objects to choose).

Step2: Substitute values into the formula

First, calculate the factorials. \( n!=15! = 15\times14\times13\times12\times11\times10! \), \( k!=5! = 5\times4\times3\times2\times1 \), and \( (n - k)!=(15 - 5)!=10! \).

Substituting into the combination formula:

$$
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Part (b)

Step1: Identify the problem type

This is a permutation problem since the order of choices is considered. The formula for permutations is \( P(n, k)=\frac{n!}{(n - k)!} \), where \( n = 15 \) and \( k = 5 \).

Step2: Substitute values into the formula

Calculate the factorials. \( n!=15! = 15\times14\times13\times12\times11\times10! \) and \( (n - k)!=(15 - 5)!=10! \).

Substituting into the permutation formula:

$$
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$$

Answer:

(a) \(\boldsymbol{3003}\)

(b) \(\boldsymbol{360360}\)