Question

For each of the following scenarios, determine if it is a permutation or combination and answer the associated questions.

1. You have to decide how many ways to put 15 players into 5 different starting positions in a sport. Complete the calculation. Change the situation so it uses the other operation (Ex. if it was a combination make it a permutation).
2. Out of 10 pairs of socks, you have to decide which 3 pairs you want to pack in your suitcase for the weekend. Complete the calculation. Change the situation so it uses the other operation (Ex. if it was a combination make it a permutation).
3. Out of 30 different songs to listen to, you have to decide how many ways can you make a playlist of 10 songs, if it doesn't matter the order the songs are played. Complete the calculation. Change the situation so it uses the other operation (Ex. if it was a combination make it a permutation).
4. You have to decide how many ways to select 3 different flavours for a 3 scoop cone of ice cream given 20 flavours of ice cream. Complete the calculation. Change the situation so it uses the other operation (Ex. if it was a combination make it a permutation).

Answer 1

The scenarios involve either permutations or combinations depending on whether the order of the items is important. Calculations for each scenario use factorial-based formulas for either permutations or combinations to find the number of ways arrangements or selections can be made.

Step-by-step explanation:

When deciding how many ways to arrange or select items, we often use permutations and combinations as mathematical concepts to help us calculate these scenarios. Permutations are used when the order matters and combinations are used when the order doesn't matter.

1. Arranging 15 players into 5 different starting positions is a permutation scenario because the order in which players are arranged matters. The calculation involves a permutation of 15 players taken 5 at a time: P(15, 5) = 15! / (15-5)! = 360,360 ways. To turn this into a combination, you could ask how many groups of 5 can be formed from 15 players without considering the order.
2. Choosing 3 pairs of socks from 10 pairs is a combination since the order in which you pack the socks doesn't matter. The calculation is the combination of 10 pairs taken 3 at a time: C(10, 3) = 10! / (3! * (10-3)!) = 120 ways. To change this situation into a permutation, you could specify that the socks have to be packed in a certain order.
3. Creating a playlist of 10 songs from 30, without caring about the order, is a combination. The calculation is C(30, 10) = 30! / (10! * (30-10)!) = 30,045,015 ways. To alter it to a permutation, you'd consider the order in which the songs are played to be important.
4. Selecting 3 different flavors for a 3 scoop cone from 20 flavors without order consideration is a combination. The calculation is C(20, 3) = 20! / (3! * (20-3)!) = 1,140 ways. To make this a permutation, specify the order in which the flavors are scooped.