Answer:
455
Step-by-step explanation:
This is a combinatorics problem.
It is asking us the number of ways a sample of r elements can be obtained from a larger set of n objects where order does not matter and repetitions are not allowed
It is called n choose k and the formula is
[tex]C(n,r) = \binom{n}{r} = \frac{n!}{( r! (n - r)! )}[/tex]
Here the ! symbol represents the factorial of that number
For example,
[tex]n! = n\cdot(n-1)\cdot(n-2)......\cdot3\cdot2\cdot 1[/tex]
Here n = 15 and r = 12
How many ways can we arrange 12 people in 15 slots
[tex]C(n,r) = C(15,12)[/tex]
[tex]= \frac{15!}{( 12! (15 - 12)! )}[/tex]
[tex]= \frac{15!}{12! \times 3! }[/tex]
[tex]= 455[/tex]
You can use a calculator to do this but noting that
15! = 15 x 14 x 13 x 12! and 3! = 3 x 2 = 6 we get
[tex]= \frac{15!}{12! \times 3! } = \frac{15\cdot14\cdot13}{6} = 455[/tex]
