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Chapter 5: Problem 41

A ski patrol unit has nine members available for duty, and two of them are to be sent to rescue an injured skier. In how many ways can two of these nine members be selected? Now suppose the order of selection is important. How many arrangements are possible in this case?

Short Answer

Expert verified
So, the combination is 36, and the permutation is 72.

Step by step solution

01

Evaluate the Combination

For the first part of the problem, we want to know how many ways we can select 2 members from a group of nine, not considering the order. This is defined as a combination. To find the number of combinations, we use the combination formula: \( C(n, k) = \frac{n!}{k!(n-k)!} \) where \( n \) is the total number of objects, \( k \) is the number of objects to be chosen, and \( ! \) denotes factorial, meaning the product of all positive integers less than or equal to that number. We plug the numbers into the formula: \( C(9, 2) = \frac{9!}{2!(9-2)!} \)
02

Calculate the Combination

Solving the expression, first calculate the factorials for the numbers. \( 9! = 9*8*7*6*5*4*3*2*1 \) and \( 2! = 2*1 \). Plug the factorials back into the formula and solve the expression to find the combinations possible for the selection of 2 members out of 9.
03

Evaluate the Permutation

For the second part of the problem, we need to find how many ways we can select and arrange 2 members from the group of 9. This is defined as a permutation. The formula for permutation is: \( P(n, k) = \frac{n!}{(n-k)!} \), which needs to be solved.
04

Calculate the Permutation

Substitute our values into the permutation formula: \( P(9, 2) = \frac{9!}{(9-2)!} \). As in Step 2, calculate the factorials for the numbers. Plug these into the formula to find the permutations possible for the selection and arrangement of 2 members out of 9.

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