the student wants to buy 3 CDs, but assume that there are 3 CDs featuring the piano, 5 CDs featuring the trumpet, and 7 CDs featuring the saxophone. (2) In how many ways can the selection be made if CD's featuring at least 2 different instruments are selected?
There are several ways you can do it, but this way is probably the easiest.
Try selecting 3 CDs from the entire set of CDs, ignoring the other parts of the problem for now. You have 15C3=455 ways.
Then find how many ways they can be chosen so that all three are the same. 3C3+5C3+7C3=46.
Subtract! 455-46 should be your answer, since there's always two different types of instruments!
This is called COMPLEMENTARY COUNTING, and it means that you are counting all possibilities then subtracting the ones you don't want. It's often much easier that way.
Well, if the student wants to buy 3 CDs featuring at least 2 different instruments, we can consider a few possibilities.
First, they could buy one piano CD and two CDs featuring the trumpet or saxophone. There are 3 ways to choose a piano CD and then 5+7=12 ways to choose the other two CDs. So that's 3*12=36 possibilities.
Second, they could buy one trumpet CD and two CDs featuring the piano or saxophone. There are 5 ways to choose a trumpet CD and then 3+7=10 ways to choose the other two CDs. So that's 5*10=50 possibilities.
Third, they could buy one saxophone CD and two CDs featuring the piano or trumpet. There are 7 ways to choose a saxophone CD and then 3+5=8 ways to choose the other two CDs. So that's 7*8=56 possibilities.
In total, there are 36+50+56=142 ways to make the selection. That's quite a few options for the student!
To solve this problem, we can use the concept of combinations.
Step 1: Calculate the number of ways to select CDs featuring only one instrument.
- Selecting 3 CDs featuring only the piano: 1 way (3 CDs featuring the piano)
- Selecting 3 CDs featuring only the trumpet: 1 way (5 CDs featuring the trumpet)
- Selecting 3 CDs featuring only the saxophone: 1 way (7 CDs featuring the saxophone)
Step 2: Calculate the number of ways to select CDs featuring exactly 2 different instruments.
- Selecting 2 CDs featuring the piano and 1 CD featuring the trumpet:
- Number of ways to select 2 CDs featuring the piano: C(3, 2) = 3
- Number of ways to select 1 CD featuring the trumpet: C(5, 1) = 5
- Total number of ways: C(3, 2) * C(5, 1) = 15
- Selecting 2 CDs featuring the piano and 1 CD featuring the saxophone:
- Number of ways to select 2 CDs featuring the piano: C(3, 2) = 3
- Number of ways to select 1 CD featuring the saxophone: C(7, 1) = 7
- Total number of ways: C(3, 2) * C(7, 1) = 21
- Selecting 2 CDs featuring the trumpet and 1 CD featuring the saxophone:
- Number of ways to select 2 CDs featuring the trumpet: C(5, 2) = 10
- Number of ways to select 1 CD featuring the saxophone: C(7, 1) = 7
- Total number of ways: C(5, 2) * C(7, 1) = 70
Step 3: Calculate the number of ways to select CDs featuring all 3 different instruments.
- Number of ways to select 1 CD each featuring the piano, trumpet, and saxophone: C(3, 1) * C(5, 1) * C(7, 1) = 105
Step 4: Sum up the results from steps 1, 2, and 3 to get the final answer.
- Total number of ways to select CDs featuring at least 2 different instruments:
1 (step 1) + 15 (step 2) + 21 (step 2) + 70 (step 2) + 105 (step 3) = 212
Therefore, there are 212 ways to make the selection if CDs featuring at least 2 different instruments are selected.
To find the number of ways to make the selection of CDs featuring at least 2 different instruments, we need to consider different cases.
Case 1: CD's featuring piano and trumpet:
In this case, we select 1 CD featuring the piano out of the available 3 CDs, and 2 CDs featuring the trumpet out of the available 5 CDs.
Number of ways = (number of ways to select piano CD) x (number of ways to select trumpet CDs)
= (3C1) x (5C2)
= 3 x (5! / (2! x 3!))
= 3 x (5 x 4 / (2 x 1))
= 60
Case 2: CD's featuring piano and saxophone:
In this case, we select 1 CD featuring the piano out of the available 3 CDs, and 2 CDs featuring the saxophone out of the available 7 CDs.
Number of ways = (number of ways to select piano CD) x (number of ways to select saxophone CDs)
= (3C1) x (7C2)
= 3 x (7! / (2! x 5!))
= 3 x (7 x 6 / (2 x 1))
= 63
Case 3: CD's featuring trumpet and saxophone:
In this case, we select 1 CD featuring the trumpet out of the available 5 CDs, and 2 CDs featuring the saxophone out of the available 7 CDs.
Number of ways = (number of ways to select trumpet CD) x (number of ways to select saxophone CDs)
= (5C1) x (7C2)
= 5 x (7! / (2! x 5!))
= 5 x (7 x 6 / (2 x 1))
= 105
Total number of ways to make the selection = Number of ways in Case 1 + Number of ways in Case 2 + Number of ways in Case 3
= 60 + 63 + 105
= 228
Therefore, there are 228 ways to make the selection if CD's featuring at least 2 different instruments are selected.