If $(1 + x)^{n} = C_{0} + C_{1} x + C_{2} x^{2} + . . . . . . . . . . . C_{n} x^{n}$ then $C_{0} . C_{r} + C_{1} . C_{r} + C_{2} . C_{r + 2} + . . . . . . . + C_{n - r} . C_{n}$

_{C_{n}}^{2 n}

No worries! We‘ve got your back. Try BYJU‘S free classes today!

^{2 n} C_{n + r}

No worries! We‘ve got your back. Try BYJU‘S free classes today!

^{2 n} C_{n - r}

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

0

No worries! We‘ve got your back. Try BYJU‘S free classes today!

Open in App

Solution

The correct option is C $^{2 n} C_{n - r}$
In some questions, substituting n=a positive number in both the question and the answer is the fastest way to achieve the correct option. Although there is always alternate options like writing the general term and splitting it to a format that can be solved but that takes long in a limited time paper. Try to put $n = 2, r = 1$ . In the end only option C remains.

Suggest Corrections

Similar questions

Q. If

c_{0}, c_{1}, c_{2}, . . . . . . . c_{n}

denote the coefficients in the expansion of

{(1 + x)}^{n}

, prove that

c_{0} c_{r} + c_{1} c_{r + 1} + c_{2} c_{r + 2} + . . . . + c_{n - r} c_{n} = \frac{| 2 n - - -}{| n - r - ---- - | n + r - ---- -}

If $(1 + x)^{n} = C_{0} + C_{1} x + C_{2} x^{2} + . . . . . . . + C_{n} x^{n}$ , then $\frac{C_{1}}{C_{0}} + \frac{2 C_{2}}{C_{1}} + \frac{3 C_{3}}{C_{2}} + . . . . . . . . + \frac{n C_{n}}{C_{n - 1}} =$

Q. For

r = 0, 1, 2, . . . ., n

, prove that

C_{0} \cdot C_{r} + C_{1} \cdot C_{r + 1} + C_{2} \cdot C_{r + 2} + . . . . + C_{n - r} \cdot C_{n} =^{2 n} C_{(n + r)}

and hence deduce that
i)

C_{0}^{2} + C_{1}^{2} + C_{2}^{2} + . . . . . . + C_{n}^{2} =^{2 n} C_{n}

ii)

C_{0} \cdot C_{1} + C_{1} \cdot C_{2} + C_{2} \cdot C_{3} + . . . . . + C_{n - 1} \cdot C_{n} =^{2 n} C_{n + 1}

Q. If

(1 + x)^{n} = C_{0} + C_{1} x + C_{2} x^{2} + \dots + C x^{2}

prove that

C_{0} + C_{1} + C_{2} + \dots + C_{n} = 2^{n}

Q. If

(1 + x)^{n} = C_{0} + C_{1} x + C_{2} x^{2} + . . . + C_{n} x^{n},

then

\frac{(C_{0} + C_{1}) (C_{1} + C_{2}) . . . (C_{n - 1} + C_{n})}{C_{1} C_{2} . . C_{n}}

equals-

Join BYJU'S Learning Program

If (1+x)n=C0+C1x+C2x2+...........Cnxn then C0.Cr+C1.Cr+C2.Cr+2+.......+Cn−r.Cn

If $(1 + x)^{n} = C_{0} + C_{1} x + C_{2} x^{2} + . . . . . . . . . . . C_{n} x^{n}$ then $C_{0} . C_{r} + C_{1} . C_{r} + C_{2} . C_{r + 2} + . . . . . . . + C_{n - r} . C_{n}$