Question

mayas digital music library has a total of 249 songs. her library has 64 songs that are remixes and 85 hip - hop songs. of the hip - hop songs in her library, 25 are remixes. how many songs in her library are neither remixes nor hip - hop songs?
a 75
b 100
c 125
d 160

Explanation:

Step1: Find the number of songs that are either remixes or hip - hop or both.

We use the principle of inclusion - exclusion. The number of remixes is \(n(R)=64\), the number of hip - hop songs is \(n(H) = 85\), and the number of songs that are both remixes and hip - hop (remix hip - hop) is \(n(R\cap H)=25\).
The formula for \(n(R\cup H)\) is \(n(R)+n(H)-n(R\cap H)\).
Substitute the values: \(n(R\cup H)=64 + 85-25\)
First, \(64+85 = 149\), then \(149 - 25=124\)? Wait, no, wait. Wait, the total number of songs is 249. Wait, maybe I made a mistake. Wait, let's re - calculate: \(64+85 = 149\), \(149-25 = 124\)? No, wait, 64+85 is 149, 149 - 25: 149-20 = 129, 129 - 5=124. Wait, but the total number of songs is 249. Then the number of songs that are neither remixes nor hip - hop is total songs minus \(n(R\cup H)\).
Wait, total songs \(N = 249\). So the number of songs that are neither is \(N - n(R\cup H)=249-(64 + 85-25)\)
First, calculate \(64 + 85-25\): \(64+85=149\), \(149 - 25 = 124\). Then \(249-124 = 125\). Wait, but let's check again.
Wait, the number of remixes is 64, hip - hop is 85, and the overlap (remix and hip - hop) is 25. So the number of unique remixes (only remixes) is \(64 - 25=39\), the number of unique hip - hop (only hip - hop) is \(85 - 25 = 60\), and the number of both is 25. Then the number of songs that are either remix or hip - hop or both is \(39+60 + 25=124\)? No, 39+60 is 99, 99+25 = 124. Then total songs is 249, so neither is \(249-124 = 125\).
Wait, let's do it step by step:

Step1: Calculate the number of songs in (remix or hip - hop)

Using inclusion - exclusion: \(n(\text{remix}\cup\text{hip - hop})=n(\text{remix})+n(\text{hip - hop})-n(\text{remix}\cap\text{hip - hop})\)
\(n(\text{remix}) = 64\), \(n(\text{hip - hop})=85\), \(n(\text{remix}\cap\text{hip - hop}) = 25\)
So \(n(\text{remix}\cup\text{hip - hop})=64 + 85-25=124\)

Step2: Calculate the number of songs that are neither

Total number of songs \(N = 249\)
Number of songs that are neither = \(N - n(\text{remix}\cup\text{hip - hop})\)
\(=249-124 = 125\)

Answer:

C. 125