Evaluate the given binomial coefficient
( 8 )
4
The given binomial coefficient is (8 choose 4).
To evaluate this coefficient, we use the formula for binomial coefficients:
(8 choose 4) = 8! / (4! * (8-4)!)
The exclamation mark denotes factorial, which means multiplying the number by all positive integers less than it down to 1.
Let's calculate:
8! = 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 = 40320
4! = 4 * 3 * 2 * 1 = 24
(8-4)! = 4! = 24
Now, substitute these values back into the formula:
(8 choose 4) = 40320 / (24 * 24) = 70
Therefore, (8 choose 4) evaluates to 70.
To evaluate the given binomial coefficient (8 choose 4), we can use the formula for binomial coefficients, which is:
(n choose k) = n! / (k! * (n - k)!)
In this case, n = 8 and k = 4.
We can calculate the factorial of 8 (8!) and the factorial of 4 (4!) to get the numerator and the denominator of the binomial coefficient.
1. Calculate 8! (factorial of 8):
8! = 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 = 40320
2. Calculate 4! (factorial of 4):
4! = 4 * 3 * 2 * 1 = 24
3. Calculate (8 choose 4):
(8 choose 4) = 40320 / (24 * (8 - 4))
= 40320 / (24 * 4)
= 40320 / 96
= 420
Therefore, the value of the binomial coefficient (8 choose 4) is 420.