UPSC NDA Math Model Paper 1

For the expression (1+x)24,n=24
(1+x)n=C0+C1x+C2x2+........ +Crxr+.......+Cn−1xn−1+Cnxn
Let's say that the successive terms at (r−1)th and rth place are the required terms.
∴

Cr

Cr−1

=

4

1

Let us simplify

Cr

Cr−1

Cr

Cr−1

=

n! r!(n−r)!

n! (r−1)![(n−(r−1)]!

=

(r−1)!(n−r+1)!

r!(n−r)!

=

n−r+1

r

∴

24−r+1

r

=

4

1

⇒24−r+1=4r
⇒5r=25
⇒r=5
∴ The required terms are at r−1=5−1=4th place and r=5th place.