a florist can select five different flowers from a selection of twelve different types. How many ways can she select these flowers?
To determine the number of ways the florist can select five different flowers from twelve different types, we can use a combination formula.
In this case, we have 12 different types of flowers, and we want to choose 5 of them.
The formula for combinations is:
C(n, r) = n! / (r! * (n - r)!)
Where:
- C(n, r) represents the number of combinations of n items taken r at a time.
- n! denotes the factorial of n, which is the product of all positive integers from 1 to n.
Using this formula, we can calculate the number of ways to select the flowers:
C(12, 5) = 12! / (5! * (12 - 5)!)
Simplifying further:
C(12, 5) = 12! / (5! * 7!)
Now let's calculate the factorial values:
12! = 12 * 11 * 10 * 9 * 8 * 7!
5! = 5 * 4 * 3 * 2 * 1
Plugging these values into the formula:
C(12, 5) = (12 * 11 * 10 * 9 * 8 * 7!) / (5 * 4 * 3 * 2 * 1 * 7!)
The 7! terms cancel out, simplifying the expression:
C(12, 5) = (12 * 11 * 10 * 9 * 8) / (5 * 4 * 3 * 2 * 1)
Now we can calculate this value:
C(12, 5) = 792
Therefore, there are 792 ways the florist can select five different flowers from the twelve different types.
To find the number of ways the florist can select five different flowers from twelve different types, we can use the concept of combination.
The formula for combination is given by:
n C r = n! / (r! * (n - r)!)
Where n is the total number of items or options, and r is the number of items to be selected.
In this case, n = 12 (the total number of flower types) and r = 5 (the number of flowers to be selected).
Plugging the values into the combination formula, we get:
12 C 5 = 12! / (5! * (12 - 5)!)
Simplifying this expression, we have:
12 C 5 = 12! / (5! * 7!)
Now let's calculate the factorial values:
12! = 12 * 11 * 10 * 9 * 8 * 7!
5! = 5 * 4 * 3 * 2 * 1
Substituting these values into the formula:
12 C 5 = (12 * 11 * 10 * 9 * 8 * 7!) / (5 * 4 * 3 * 2 * 1 * 7!)
The 7! terms cancel out, leaving us with:
12 C 5 = (12 * 11 * 10 * 9 * 8) / (5 * 4 * 3 * 2 * 1)
Calculating this expression, we get:
12 C 5 = 792
Therefore, there are 792 ways the florist can select five different flowers from twelve different types.