A florist has access to 9 types of flowers and 8 types of foliage. He makes bouquets using 4 types of flowers and 3 types of foliage. How many different bouquets can he make?
For flowers, it's out of 9 choose 4, or C(9,4). For foliage, it's 8 choose 3, or C(8,3).
The formula for combinations, C(n,r) is n!/(r!(n-r)!).
For flowers, C(9,4)=9!/(4!5!)=9*8*7*6/(1*2*3*4)=126.
Calculate similarly C(8,3) and multiply the two results together to get the number of different bouquets.
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To determine the number of different bouquets the florist can make, we need to calculate the number of combinations of flowers and foliage.
First, let's find the number of ways to choose 4 types of flowers out of the 9 available options. We can calculate this using the combination formula:
C(n, r) = n! / (r!(n-r)!)
In this case, n = 9 (number of available flower types) and r = 4 (number of flowers to choose).
C(9, 4) = 9! / (4!(9-4)!)
= 9! / (4!5!)
= (9 * 8 * 7 * 6) / (4 * 3 * 2 * 1)
= 9 * 2 * 7
= 126
So there are 126 different ways to choose 4 types of flowers from the given 9 types.
Next, let's find the number of ways to choose 3 types of foliage out of the 8 available options. Using the same combination formula:
C(8, 3) = 8! / (3!(8-3)!)
= 8! / (3!5!)
= (8 * 7 * 6) / (3 * 2 * 1)
= 56
There are 56 different ways to choose 3 types of foliage from the given 8 types.
To get the total number of different bouquets, we multiply the number of ways to choose flowers and foliage:
Total number of bouquets = 126 * 56
= 7056
Therefore, the florist can make 7056 different bouquets using the given set of flowers and foliage.