A student wants to buy 3 CDs, but assume that there are 6 CDs featuring the piano,
4 CDs featuring the trumpet, and
7 CDs featuring the saxophone. In how many ways can the selection be made if
CD's featuring at least 2 different instruments
are selected?
We want selections of the kind
PP?
SS?
TT?
1. 2 piano, one non-piano --> C(6,2)*C(11,1) = 15(11) = 165
2. 2 trumpet, one non-trumpet --> C(4,2)*(C(13,1) = 78
3. 3 sax, one non-sax --> C(7,2)*C(10,1) = 210
number of selections = 165+78+210 = 453
453
wrong answer.
To find the number of ways the student can select CDs featuring at least 2 different instruments, we can use the concept of combinatorics.
First, we need to consider the different combinations of instruments that can be selected.
Since there are 6 CDs featuring the piano, 4 CDs featuring the trumpet, and 7 CDs featuring the saxophone, we have:
- CD combinations with only the piano and trumpet: We can select 3 CDs from the 6 piano CDs and 4 CDs from the 4 trumpet CDs. This can be calculated using the combination formula: C(6,3) * C(4,4) = 20 * 1 = 20.
- CD combinations with only the piano and saxophone: We can select 3 CDs from the 6 piano CDs and 4 CDs from the 7 saxophone CDs. This can be calculated using the combination formula: C(6,3) * C(7,4) = 20 * 35 = 700.
- CD combinations with only the trumpet and saxophone: We can select 3 CDs from the 4 trumpet CDs and 4 CDs from the 7 saxophone CDs. This can be calculated using the combination formula: C(4,3) * C(7,4) = 4 * 35 = 140.
- CD combinations with all three instruments: We can select 3 CDs from the 6 piano CDs, 4 CDs from the 4 trumpet CDs, and 4 CDs from the 7 saxophone CDs. This can be calculated using the combination formula: C(6,3) * C(4,4) * C(7,4) = 20 * 1 * 35 = 700.
To find the total number of ways the selection can be made, we need to sum up all the above combinations:
20 + 700 + 140 + 700 = 1560.
Therefore, there are 1560 ways the student can select CDs featuring at least 2 different instruments.