Question

Which of the following terms represents the value of n in the expansion of (1+x)^n such that the coefficients of the terms containing x^5and x^12 are equal?

a) 7
b) 17
c) 8
d) 12

Answer 1

To find the value of n in the expansion of (1+x)^n such that the coefficients of the terms containing x^5 and x^12 are equal, we can use the binomial theorem. By equating the coefficients and solving the resulting equation, we find that the value of n is 15.

Step-by-step explanation:

The coefficients of the terms in the expansion of (1+x)^n can be determined using the binomial theorem. By comparing the coefficients of x^5 and x^12, we can set up two equations and solve for n. In the expansion, the coefficient of x^5 is given by n!/(5!(n-5)!), and the coefficient of x^12 is given by n!/(12!(n-12)!). Equalizing these coefficients, we have:

n!/(5!(n-5)!) = n!/(12!(n-12)!)

Cancelling out the factorial terms, we get:

(n-5)! = (12!(n-12)!)/(5!)

By simplifying further, we find:

(n-5)! = (12*11*10*9*8*7*(n-12)!)/(5*4*3*2*1)

Cancelling out the (n-12)! terms, we have:

(n-5)! = (12*11*10*9*8*7)/(5*4*3*2*1)

Now, we can list out the factors of (12*11*10*9*8*7) and check which factorials contain the factor of 5. From the list of factors, we can see that the only factorials that contain the factor of 5 are (10!) and (5!), which means that n-5 = 10. Therefore, n = 15.