A disc jockey has 12 songs to play. Seven are slow songs, and five are fast songs. Each song is to be played only once. In how many ways can the disc jockey play the 12 songs if
The songs can be played in any order.
The first song must be a slow song and the last song must be a slow song.
The first two songs must be fast songs.
WHAT IN THE WORLD DO I DO??
The first thing you need to do is proofread your post.
You state that the first song must be a slow song. Yet in the next sentence you say that the first two song must be fast.
Which is it?
Well its saying IF:
*The songs can be played in any order.
*The first song must be a slow song and the last song must be a slow song.
*The first two songs must be fast songs.
Jen -- does that make sense to you??
Ohh -- I see (I think). You need to figure the order two ways.
how do i do this??
To answer these types of counting questions, we can use the concept of permutations. We can solve each scenario separately:
1. The songs can be played in any order:
In this case, we have 12 songs to play, so we have 12 options for the first song, then 11 options for the second song, and so on. Therefore, the number of ways to play the songs in any order is 12!
2. The first song must be a slow song and the last song must be a slow song:
In this case, we have 7 slow songs that can be chosen as the first song, and 6 slow songs left that can be chosen as the last song. Then we have 10 remaining songs to be chosen for the remaining positions. Therefore, the number of ways to arrange the songs with the first and last song being slow is 7 * 6 * 10!.
3. The first two songs must be fast songs:
In this case, we have 5 fast songs to choose from for the first song. After selecting the first fast song, we have 4 remaining fast songs to choose from for the second song. Then we have 10 remaining songs to be chosen for the remaining positions. Therefore, the number of ways to arrange the songs with the first two songs being fast is 5 * 4 * 10!.
To find the total number of ways, we need to multiply the results from each scenario together. So, the total number of ways the disc jockey can play the 12 songs satisfying all the conditions is (7 * 6 * 10!) * (5 * 4 * 10!) * 12!.
Note: "!" represents the factorial operator, which means multiplying all the positive integers up to a given number. For example, 5! = 5 * 4 * 3 * 2 * 1 = 120.