Problem 4 - A First Course in Probability

Question

John, Jim, Jay, and Jack have formed a band consisting of 4 instruments. If each of the boys can play all 4 instruments, how many different arrangements are possible? What if John and Jim can play all 4 instruments, but Jay and Jack can each play only piano and drums?

Step-by-Step Solution

Answer

There are 24 possible arrangements when all boys can play all 4 instruments, and 18 different arrangements when John and Jim can play all instruments while Jay and Jack can only play piano and drums.

Step 1: Part 1: All boys can play all 4 instruments

In this case, we will use permutation formula for arranging the 4 instruments among the 4 band members where each instrument can only be played by one boy at a time. The permutation formula is: n! / (n-r)! where n is the number of items, r is the number of positions the items can take and n! represents the factorial of n. In our case, n = 4 (since there are 4 instruments) and r = 4 (since each instrument is assigned to one of the 4 boys). So, 4! / (4-4)! = 4! / 0! = 4! / 1 = 4 × 3 × 2 × 1 = 24 So, there are 24 different arrangements possible when each boy can play all 4 instruments.

Step 2: Part 2: John and Jim can play all 4 instruments, but Jay and Jack can only play piano and drums

Now we have to take into consideration that each person can play a different number of instruments. Let's break it down: 1. If John or Jim plays the piano, there are 2 options for this. Then, the other one can play one of the 3 remaining instruments (guitar, bass, or drums). After that, Jay and Jack can only play drums, which is 1 option. So, we have 2 x 3 x 1 = 6 arrangements in this case. 2. If Jay plays the piano, there's only 1 option. Then, John or Jim can play one of the 3 remaining instruments. Since there are only two possible people here, there are 2 x 3 = 6 arrangements. 3. If Jack plays the piano, there's only 1 option. Then, John or Jim can play one of the 3 remaining instruments. On the same logic as before, there are 2 x 3 = 6 arrangements. Summing the total arrangements from the 3 cases above, we have 6 + 6 + 6 = 18 different arrangements possible when considering the limitations of Jay and Jack. Thus, there are: - 24 different arrangements where all boys can play all 4 instruments - 18 different arrangements where John and Jim can play all instruments and Jay and Jack can play only the piano and drums.

Key Concepts

Understanding Factorial

Factorial is a mathematical concept that helps us find the number of ways to arrange a set of items. It's represented as an exclamation mark (!).
For example, the factorial of 4 is written as 4! and means multiplying all whole numbers from 4 down to 1:

• 4! = 4 × 3 × 2 × 1 = 24

This means there are 24 ways to arrange 4 items. Factorials are used heavily in permutations to calculate possible arrangements. If you're dealing with arranging items, factorials are your go-to tool! Keep in mind, the factorial of 0 is 1, which often appears in calculations.

Arrangements Made Simple

Arrangements involve organizing or ordering a set of items. For example, arranging 4 musical instruments among 4 boys means deciding who plays what.
When all possibilities are open, like in part one of our exercise, any of the boys can play any instrument. This creates tons of possible setups!
If each boy can play any of the instruments, the total number of arrangements is 24, as we found using the factorial concept. Think of arrangements as varieties of possible setups or combinations. They change when conditions, like skills or preferences, are added.

The Permutation Formula

Permutations refer to the different ways to arrange a set of items. The permutation formula helps us calculate this:

• \[ P(n, r) = \frac{n!}{(n-r)!} \]

Where:

• \( n \) is the total number of items.
• \( r \) is the number of positions to fill.

In our exercise, we used it to find how many ways 4 instruments can be split among 4 boys. Since each boy gets one instrument, \( n = r = 4 \), so the formula gives us 24 arrangements.
Permutations change with the number of items and positions, allowing for flexible problem solving.

Musical Instruments Arrangement

Arranging musical instruments among band members highlights how permutations can change based on conditions. Initially, with no restrictions, each boy playing any instrument gives us 24 arrangements.
But when skills limit who can play what, like Jay and Jack only handling drums and piano, it reduces possibilities.
For this scenario, breaking down options by each case clarified possible setups:

• When John or Jim plays piano, we have 6 new configurations.
• Similarly, when Jay or Jack plays, 6 variations emerge too.

This shifts the total to 18 distinct arrangements. The shift demonstrates how real-life constraints complicate arrangements, making permutation logic crucial in decision-making.