Final answer:
The ninth term of the binomial expansion (p+2r)^11 is calculated using the binomial theorem, which results in the term 165*p^3*256r^8.
Step-by-step explanation:
To find the ninth term of the binomial expression (p+2r)^11, we can use the binomial theorem. The general form of the binomial theorem for any positive integer n and any numbers a and b is given as:
(a + b)^n = a^n + nC1*a^(n-1)*b^1 + nC2*a^(n-2)*b^2 + ... + nC(n-1)*a^1*b^(n-1) + b^n
where "nCk" represents the binomial coefficient, equivalent to n!/(k!(n-k)!).
The r-th term of the binomial expansion (a + b)^n is given by:
T(r) = nC(r-1)*a^(n-r+1)*b^(r-1)
For the ninth term of (p+2r)^11, r is 9. Plugging the values into the formula, we get:
T(9) = 11C8*p^3*(2r)^8
Which simplifies to
T(9) = 11C8*p^3*256r^8
Next, we calculate the binomial coefficient 11C8:
11C8 = 11!/(8!*(11-8)!)
= 11!/(8!*3!)
= (11*10*9)/(3*2*1)
= 165
Now we can complete our calculation:
T(9) = 165*p^3*256r^8
Thus, the ninth term of the binomial expression (p+2r)^11 is 165*p^3*256r^8.