Simplify 8C5
A 6,720
B 336
C 56
D 10
pls helpp
The factorial definition is great, but I have often found it convenient that to find nCr, note two things.
(1) if r > n/2, then ncR = nC(n-r)
For example, 20C17 = 20C3
This makes it easier to evaluate
(2) write the fraction by counting down from n in the numerator, and counting up from 1 in the denominator. Count for r factors.
20C3 = 20*19*18 / 1*2*3 = 20*19*6*3 / 1*2*3 = 20*19*3
It is easy to split out factors that cancel out in the denominator
Note that 20C17 = 20*19*18*17*16*...*4 / 1*2*3*4*...*17
Note that the last 14 factors cancel directly, showing that 20C17 = 20C3
To simplify 8C5, we need to calculate the number of ways to choose 5 items from a set of 8 items without considering the order.
The formula for calculating combinations is given by:
nCk = n! / (k! * (n - k)!)
where n is the total number of items and k is the number of items to be chosen. "!" denotes the factorial of a number, which means multiplying all positive integers from 1 up to that number.
So to calculate 8C5:
8C5 = 8! / (5! * (8 - 5)!)
Now, let's calculate the factorials and simplify the expression:
8! = 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1
5! = 5 * 4 * 3 * 2 * 1
(8 - 5)! = 3! = 3 * 2 * 1
Substituting these values back into the expression:
8C5 = (8 * 7 * 6 * 5 * 4 * 3 * 2 * 1) / ((5 * 4 * 3 * 2 * 1) * (3 * 2 * 1))
Now, simplify the expression further:
8C5 = (8 * 7 * 6) / (3 * 2 * 1)
8C5 = 336/6
Finally, perform the division:
8C5 = 56
Therefore, the answer is C) 56.