Problem 47
Question
Nine bands have volunteered to perform at a benefit concert, but there is only enough time for five of the bands to play. How many lineups are possible?
Step-by-Step Solution
Verified Answer
The number of possible lineups is 15120.
1Step 1: Understanding The Problem
Firstly, one must understand the problem and note that this is a situation of permutation where order matters because the performances by bands are to occur one after the other, which makes the order of importance.
2Step 2: Identify The Formula
The formula for permutations of n items taken r at a time is given by \( nPr = \frac{n!}{(n-r)!} \)
3Step 3: Apply The Formula
Now, apply the formula for permutation. Here, n = 9 (the total number of bands) and r = 5 (the number of bands required to perform). So, the number of ways is given by \( 9P5 = \frac{9!}{(9-5)!} \)
4Step 4: Calculate The Number of ways
By calculating the factorials and making the division gives the total possible lineups = 15120
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