Martin's iPod has 24 songs on it. He wants to make a playlist consisting of three songs to dance to. Let N be the number of different playlists Martin can make. What are the last 3 digits of N?
number of playlists
= C(24,3) = 2024
Why the last 3 digits?
What a strange question.
To find the number of different playlists Martin can make, we need to use combinations.
Since Martin wants to make a playlist of three songs out of the 24 songs on his iPod, we need to calculate the combination or "24 choose 3".
Combination formula: nCk = n! / (k! * (n-k)!)
Using this formula, we can calculate:
24C3 = 24! / (3! * (24-3)!)
Simplifying:
24! = 24 * 23 * 22 * ... * 2 * 1
3! = 3 * 2 * 1
(24-3)! = 21!
24! = 24 * 23 * 22 * 21!
Now we substitute the values:
24C3 = (24 * 23 * 22) / (3 * 2 * 1) * (21!)
Calculating the first part:
24 * 23 * 22 = 12144
Let's simplify the expression further:
24C3 = 12144 / (3 * 2 * 1) * (21!)
Now we calculate (21!):
21! = 21 * 20 * 19 * ... * 2 * 1
Since we only need to find the last 3 digits of N, we can simplify the expression by calculating the factorial modulo 1000.
21! % 1000 = (21 * 20 * 19 * ... * 2 * 1) % 1000
Calculating the second part modulo 1000 might be time-consuming. To simplify the calculation, we can break down 21! into smaller parts and calculate each part modulo 1000.
21! % 1000 = (21 * 20 * 19 * ... * 2 * 1) % 1000
= ((20 % 1000) * (19 % 1000) * ... * (2 % 1000) * (1 % 1000)) % 1000
Calculating each part modulo 1000:
20 % 1000 = 20
19 % 1000 = 19
.
.
.
2 % 1000 = 2
1 % 1000 = 1
Let's substitute these values back into the expression:
21! % 1000 = (20 * 19 * ... * 2 * 1) % 1000
Now, to find the last 3 digits of N, we simplify:
24C3 = (12144 / (3 * 2 * 1) * ((20 * 19 * ... * 2 * 1) % 1000)
Calculating the first part:
12144 / (3 * 2 * 1) = 2024
Substituting back into the expression:
24C3 = 2024 * ((20 * 19 * ... * 2 * 1) % 1000)
Now, we calculate ((20 * 19 * ... * 2 * 1) % 1000) using a calculator or programming language.
After calculating this value, we can then multiply it by 2024 to find the last 3 digits of N.