An album has 15 songs. You make a playlist by randomly shuffling the order of the songs. Which values represent the probability that the first 3 songs in the playlist are the first 3 songs on the album in any order?Responses$\frac{1}{_{15}C_3}$115C3the fraction with numerator 1 and denominator 15 cap c sub 3$\frac{1}{_{15}P_3}$115P3the fraction with numerator 1 and denominator 15 cap p sub 3$\frac{1}{455}$14551 over 455$\frac{1}{2730}$12730
Question
An album has 15 songs. You make a playlist by randomly shuffling the order of the songs. Which values represent the probability that the first 3 songs in the playlist are the first 3 songs on the album in any order?Responses115C3the fraction with numerator 1 and denominator 15 cap c sub 3115P3the fraction with numerator 1 and denominator 15 cap p sub 314551 over 45512730
Solution
The correct answer is , which is the fraction with numerator 1 and denominator 15 cap p sub 3.
Here's why:
The problem is asking for the probability that the first 3 songs in the playlist are the first 3 songs on the album in any order. This is a permutation problem because the order of the songs matters.
In permutations, the order of selection is important. The formula for permutations is nPr = n! / (n - r)!, where n is the total number of items, and r is the number of items to choose.
In this case, n = 15 (the total number of songs on the album) and r = 3 (the number of songs we're interested in).
So, the total number of ways to arrange the 15 songs is 15P3 = 15! / (15 - 3)! = 2730.
The event we're interested in (the first 3 songs being the first 3 songs on the album in any order) can occur in 3P3 = 3! / (3 - 3)! = 6 ways.
So, the probability of this event is 6 / 2730 = 1 / 455.
However, the question asks for the reciprocal of this (the number of total outcomes over the number of successful outcomes), so the correct answer is 1 / (1 / 455) = 455, which is represented as .
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