Tom's new CD player has a 9-disc changer. If she owns 45 CDs, in how many ways can he choose 9 CDs to listen to?
After picking 9 CDs, in how many ways can he order them in the 9 slots of the changer?
45 items, taken 9 at a time
C(45,9)=45!/9!(45-9)!
orders: 9!
To find the number of ways Tom can choose 9 CDs to listen to, we can use the concept of combinations. The formula for calculating combinations is:
C(n, r) = n! / (r! * (n-r)!)
Where n is the total number of CDs and r is the number of CDs to be chosen.
In this case, Tom has 45 CDs and wants to choose 9 of them. So the number of ways he can choose 9 CDs can be calculated as:
C(45, 9) = 45! / (9! * (45-9)!)
Now let's calculate this value:
C(45, 9) = 45! / (9! * 36!)
= (45 * 44 * 43 * 42 * 41 * 40 * 39 * 38 * 37) / (9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1)
To calculate this expression, we can use a calculator or simplify the numerator and denominator separately.
After picking 9 CDs, the number of ways Tom can order them in the 9 slots of the changer can be calculated using permutations. The formula for calculating permutations is:
P(n, r) = n! / (n - r)!
Where n is the total number of CDs chosen and r is the number of slots to be filled.
In this case, Tom has already chosen 9 CDs, so the number of ways he can order them in the 9 slots can be calculated as:
P(9, 9) = 9! / (9 - 9)!
= 9! / 0!
Since any number divided by zero is undefined, the number of ways to order the 9 CDs in the slots is simply 1, as there is only one arrangement possible when all CDs are selected.
Therefore, Tom can choose 9 CDs from his collection in C(45, 9) ways, and he can order them in the 9 slots of the changer in 1 way.