Question

Suppose we want to choose 5 objects, without replacement, from 17 distinct objects. (a) How many ways can this be done, if the order of the choices is relevant? (b) How many ways can this be done, if the order of the choices is not relevant?

Answer 1

To determine the number of ways to choose 5 objects from 17 distinct objects, we use permutations (17! / 12!) when the order is relevant, and combinations (17! / (5! * 12!)) when order is not relevant.

Step-by-step explanation:

Calculating Combinations and Permutations

To answer the student's question:

(a) When the order of the choices is relevant, we use permutations to calculate the number of ways to choose 5 objects from 17 distinct objects. The formula for permutations is P(n, r) = n! / (n-r)!, where n is the total number of objects to choose from, and r is the number of objects we are choosing. In this case, P(17, 5) = 17! / (17-5)! = 17! / 12! which calculates the possible arrangements.

(b) When the order of the choices is not relevant, we use combinations to determine the number of ways to choose 5 objects from 17. The formula for combinations is C(n, r) = n! / [r! * (n-r)!]. So, C(17, 5) = 17! / (5! * (17-5)!) computes the number of ways we can choose the objects without considering the order.

Answer 2

The number of ways to choose 5 objects from 17 with order being relevant is calculated using permutations, while the number of ways without considering order is calculated using combinations.

Step-by-step explanation:

To answer the student's question, we need to apply concepts of combinatorics. If order is relevant (the permutation case), the number of ways to choose 5 objects from 17 distinct objects is calculated using the formula for permutations of n objects taken r at a time, which is P(n, r) = n! / (n-r)! For our case, this would be P(17, 5) = 17! / (17-5)! = 17! / 12!.

If order is not relevant (the combination case), the number of ways to choose 5 objects from 17 without regard to order is calculated using the formula for combinations of n objects taken r at a time, which is C(n, r) = n! / [r! (n-r)!].

For our case, this would be C(17, 5) = 17! / [5! (17-5)!] = 17! / [5! 12!].