A student wants to buy 3 CDs, but assume that there are 3 CDs featuring the piano, 5 CDs featuring the trumpet, and 3 CDs featuring the saxophone. In how many ways can the selection be made if CD's featuring at least 2 different instruments are selected?
So we can't have
TTT
PPP
SSS
could have
PST --- C(3,1) x C(3,1) x C(5,1) = 4
PPT --- C(3,2) x C(5,1) = 15
PPS --- C(3,2) x C(3,1) = 9
TTP --- C(5,2) x C(3,1) = 30
TTS --- C(5,2) x C(3,1) = 30
SSP --- C(3,2) x C(3,1) = 9
SST --- C(3,2) x C(5,1) = 15
total = 108
108 is not correct. Thanks for trying tho Reiny!
It was a simple typo,
in the first of the listings , it should have been
PST --- C(3,1) x C(3,1) x C(5,1) = 45
changing my total to 153
To calculate the number of ways to select CDs featuring at least 2 different instruments, we can use the principle of inclusion-exclusion.
First, let's determine the total number of ways to select 3 CDs:
The student can choose any CD from the 11 available options for the first selection, then any CD from the remaining 10 options for the second selection, and finally any CD from the remaining 9 options for the third selection. So the total number of ways to select 3 CDs without any restrictions is:
11 * 10 * 9 = 990 ways
Now let's calculate the number of ways to select CDs featuring only one instrument:
- CD's featuring only the piano: There are 3 options for this selection.
- CD's featuring only the trumpet: There are 5 options for this selection.
- CD's featuring only the saxophone: There are 3 options for this selection.
To calculate the number of ways to select CDs featuring only one instrument, we add these values together:
3 + 5 + 3 = 11 ways
However, we only want to count the cases where at least 2 different instruments are selected. So we need to subtract this count from the total count calculated earlier.
Total ways to select 3 CDs - Ways to select CDs featuring only one instrument = 990 - 11 = 979 ways
Therefore, there are 979 ways to make the selection if CDs featuring at least 2 different instruments are chosen.