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Chapter 4: Problem 89

An ice cream shop offers 25 flavors of ice cream. How many ways are there to select 2 different flavors from these 25 flavors? How many permutations are possible?

Short Answer

Expert verified
There are 300 different combinations and 600 permutations of ice cream flavors.

Step by step solution

01

Calculate Combinations

We can use the combination formula to calculate the total number of ways to choose 2 flavors out of 25. The combination formula is \[ C(n, k) = n! / [k!(n-k)!] \]. Here, 'n' is the total number of items available, 'k' is the number of items to select, and '!' denotes factorial. Substituting 'n' with 25 and 'k' with 2, we get \[ C(25, 2) = 25! / [2!(25-2)!] \]
02

Calculate Factorial and Substitution

To calculate the expression above we first need to calculate the values of 25!, 2! and (25-2)!. 25! is the multiplication of all positive integers up to 25, 2! is the multiplication of all positive integers up to 2, and (25-2)! is the multiplication of all positive integers up to (25-2)=23. When we do that we find these values and can substitute them into the formula, getting \[ C(25, 2) = 25*24 / [2*1] = 300 \] This means there are 300 combinations.
03

Calculate Permutations

Permutations take into account the order of selection. We can use the permutation formula, which is \[ P(n, k) = n! / (n-k)! \] Substituting 'n' with 25 and 'k' with 2, we get \[ P(25, 2) = 25! / (25-2)! \]
04

Calculate Factorial and Substitution for Permutations

Similar to what was done for combinations, we need to calculate the values of 25! and (25-2)!. After calculating these and substituting them into the formula, we get \[ P(25, 2) = 25*24 = 600 \] This means there are 600 permutations.

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