Final answer:
To find C(10,r), set up an equation using the given information. Solve the equation to find the value of r. Plug the value of r back into the formula for combinations to find C(10,r) and P(10,r).
Step-by-step explanation:
To find C(10,r), we first need to find the value of r. Since C(20,r) = C(20, r+2), we can set up an equation:
C(20,r) = C(20, r+2)
To solve this equation, we can use the formula for combinations: C(n,r) = n! / (r!(n-r)!)
Plugging in the values and simplifying, we get:
20! / (r!(20-r)!) = 20! / ((r+2)!(18-r)!)
Canceling out the common terms, we can simplify further:
1 / (r!(20-r)!) = 1 / ((r+2)!(18-r)!)
To solve for r, we can cross-multiply and solve for r:
(r+2)!(18-r)! = (r!(20-r)!)
(r+2)(r+1)(18-r)! = 20(19)(r)(r-1)(r-2)!
Expanding and simplifying, we get:
r(r+2)(r+1)(18-r)(17-r) = 380(r)(r-1)(r-2)!
Simplifying further, we get:
(r+2)(r+1)(18-r)(17-r) = 380(r-2)!
After canceling out the common terms, we are left with:
(r+2)(r+1)(18-r)(17-r) = 380
We can solve this equation algebraically to find the value of r. Once we know the value of r, we can plug it back into the formula for combinations to find C(10,r) and P(10,r).