A student wants to buy 3 CDs. Assume that they are interested in 3 CDs featuring the piano, 6 CDs featuring the trumpet, and 5 CDs featuring the saxophone.

Question

(1) In how many different ways can the 3 CD's be selected?

(2) In how many ways can the selection be made if CD's featuring at least 2 different instruments are selected?

Answer 1

In this problem, the order does not matter so you are looking for all possible combinations. I recommend using an nCr calculator or if you know the equation, it is Combinations, C = (n!) / (r!( n! - r!)), where n is the total number of options and r is the number of selections.

1) In this scenario, you can select 3 CDs from an available 14 available, so r = 3 and n = 14. Using the equation or calculator, you get 364.

2) For this one you can take your total combinations and subtract out all combinations where you have 3 of all the same CD type (i.e., you don't have at least 2 CDs of different types since they are all the same).

Looking at the Piano CDs - you can quickly realize there is only one scenario where you have all 3 Piano CDs since there are only 3. You can use the calculator to confirm this where both n and r are equal to 3.

Looking at the Trumpet CDs, n will be 6 and r is again 3. Using the calculator or equation, you will get 20 combinations.

Similarly for Saxophone CDs, use the calculator where n =5 and r = 3. You will get 10 combinations.

Now you can subtract these three sets of combinations from the total possible combinations to get:

364 - 1 - 20 - 10 = 333.