Simplify 15C3
15C3 can be simplified as:
15C3 = (15!)/((15-3)!3!)
= (15 x 14 x 13 x 12!)/(12! x 3 x 2 x 1)
= (15 x 14 x 13)/(3 x 2 x 1)
= 455
Therefore, 15C3 simplifies to 455.
To simplify 15C3, we need to evaluate the binomial coefficient expression. The formula for the binomial coefficient is:
nCr = n! / (r!(n - r)!)
In this case, we have n = 15 and r = 3, so we can substitute these values into the formula:
15C3 = 15! / (3!(15 - 3)!)
Now, let's simplify the expression:
15! = 15 x 14 x 13 x 12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1
3! = 3 x 2 x 1
(15 - 3)! = 12! = 12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1
Substituting these values into the formula:
15C3 = (15 x 14 x 13 x 12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1) / ((3 x 2 x 1) x (12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1))
Now we can simplify the expression further by canceling out common factors:
15C3 = (15 x 14 x 13) / (3 x 2 x 1)
Evaluating the expression:
15C3 = 455
To simplify 15C3, we need to evaluate the combination formula:
nCr = n! / (r! * (n-r)!)
In this case, n (the total number of items) is 15, and r (the number of items chosen) is 3.
Let's calculate each part step by step:
1. Calculate n!:
n! = 15! = 15 x 14 x 13 x 12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1
2. Calculate r!:
r! = 3! = 3 x 2 x 1
3. Calculate (n-r)!:
(n-r)! = (15-3)! = 12! = 12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1
Now, we plug these values into the combination formula:
15C3 = 15! / (3! * (15-3)!)
= (15 x 14 x 13 x 12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1) / (3 x 2 x 1) x (12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1)
By multiplying and canceling out common terms, we get:
15C3 = (15 x 14 x 13) / (3 x 2 x 1)
= 455
Therefore, 15C3 simplifies to 455.