Properties of Binomial Coefficients

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JEE Main 2015

The sum of coefficients of integral power of $x$ in the binomial expansion $(1 - 2 x)^{50}$ is

\frac{1}{2} (3^{50} - 1)

(B)

\frac{1}{2} (2^{50} + 1)

(C)

\frac{1}{2} (3^{50} + 1)

(D)

\frac{1}{2} (3^{50})

JEE Main 2010

LEVELJEE Main

Let $S_{1} = \sum_{j = 1}^{10} j (j - 1)^{10} C_{j}, S_{2} = \sum_{j = 1}^{10} j^{10} C_{j}$ and $S_3 = \sum_{j=1}^{10} j^2 ^{10}C_j$ . Statement-1: $S_{3} = 55 \times 2^{9}$ . Statement-2: $S_{1} = 90 \times 2^{8}$ and $S_{2} = 10 \times 2^{8}$ .

(A)

Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1

(B)

Statement-1 is true, Statement-2 is false.

(C)

Statement-1 is false, Statement-2 is true.

(D)

Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.

JEE Main 2008

LEVELJEE Main

Statement-1: $\sum_{r = 0}^{n} (r + 1)^{n} C_{r} = (n + 2) 2^{n - 1}$ . Statement-2: $\sum_{r = 0}^{n} (r + 1)^{n} C_{r} x^{r} = (1 + x)^{n} + n x (1 + x)^{n - 1}$ .

(A)

Statement-1 is false, Statement-2 is true

(B)

Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1

(C)

Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1

(D)

Statement-1 is true, Statement-2 is false

JEE Main 2007

LEVELJEE Main

The sum of the series $^{20} C_{0} -^{20} C_{1} +^{20} C_{2} -^{20} C_{3} + .... +^{20} C_{10}$ is

(A)

(B)

^{20} C_{10}

(C)

-^{20} C_{10}

(D)

\frac{1}{2}^{20} C_{10}

JEE Main 2005

LEVELBoard

The value of $^{50}C_4 + \sum_{r=1}^6 ^{56-r}C_3$ is

(A)

^{55} C_{4}

(B)

^{55} C_{3}

(C)

^{56} C_{3}

(D)

^{56} C_{4}

JEE Main 2005

LEVELJEE Main

If the coefficients of $r$ th, $(r + 1)$ th, and $(r + 2)$ th terms in the binomial expansion of $(1 + y)^{m}$ are in A.P., then $m$ and $r$ satisfy the equation

(A)

m^{2} - m (4 r - 1) + 4 r^{2} - 2 = 0

(B)

m^{2} - m (4 r + 1) + 4 r^{2} + 2 = 0

(C)

m^{2} - m (4 r + 1) + 4 r^{2} - 2 = 0

(D)

m^{2} - m (4 r - 1) + 4 r^{2} + 2 = 0

JEE Main 2004

LEVELBoard

If $S_{n} = \sum_{r = 0}^{n} \frac{1}{^{n} C _{r}}$ and $t_{n} = \sum_{r = 0}^{n} \frac{r}{^{n} C _{r}}$ , then $\frac{t _{n}}{S _{n}}$ is equal to

(A)

\frac{2 n - 1}{2}

(B)

\frac{n}{2} - 1

(C)

n - 1

(D)

\frac{n}{2}

JEE Main 2003

LEVELJEE Main

If $f (x) = x^{n}$ , then the value of $f (1) - \frac{f ^{'} ( 1 )}{1 !} + \frac{f ^{''} ( 1 )}{2 !} - \frac{f ^{'''} ( 1 )}{3 !} + \dots + \frac{( - 1 ) ^{n} f ^{n} ( 1 )}{n !}$ is

(A)

1

(B)

2^{n}

(C)

2^{n} - 1

(D)

0

JEE Main 2002

LEVELBoard

The coefficients of $x^{p}$ and $x^{q}$ in the expansion of $(1 + x)^{p + q}$ are

(A)

equal

(B)

equal with opposite signs

(C)

reciprocals of each other

If the sum of the coefficients in the expansion of $(a + b)^{n}$ is 4096, then the greatest coefficient in the expansion is

$r$ and $n$ are positive integers $r > 1, n > 2$ and coefficient of $(r + 2)^{t h}$ term and $3 r^{t h}$ term in the expansion of $(1 + x)^{2 n}$ are equal, then $n$ equals

(A)

3 r

(B)

3 r + 1

(C)

2 r

(D)

2 r + 1

JEE Advanced 2010

LEVELJEE Main

For $r = 0, 1, ..., 10$ , let $A_{r}, B_{r}$ and $C_{r}$ denote, respectively, the coefficient of $x^{r}$ in the expansions of $(1 + x)^{10}, (1 + x)^{20}$ and $(1 + x)^{30}$ . Then $\sum_{r = 1}^{10} A_{r} (B_{10} B_{r} - C_{10} A_{r})$ is equal to

(A)

B_{10} - C_{10}

(B)

A_{10} (B_{10}^{2} - C_{10} A_{10})

(C)

(D)

C_{10} - B_{10}

JEE Advanced 2005

LEVELJEE Main

The value of $(0 30) (10 30) - (1 30) (11 30) + (2 30) (12 30) - .... + (20 30) (30 30)$ is

(A)

(10 30)

(B)

(15 30)

(C)

(30 60)

(D)

(10 31)

JEE Advanced 2002

LEVELJEE Main

The sum $\sum_{i = 0}^{m} (i 10) (m - i 20)$ , (where $(q p) = 0$ if $p < q$ ) is maximum when $m$ is

For $2 \leq r \leq n$ , $(r n) + 2 (r - 1 n) + (r - 2 n) =$

(A)

(r - 1 n + 1)

(B)

2 (r + 1 n + 1)

(C)

2 (r n + 2)

(D)

(r n + 2)

JEE Advanced 1998

LEVELBoard

If $a_{n} = \sum_{r = 0}^{n} \frac{1}{^{n} C _{r}}$ , then $\sum_{r = 0}^{n} \frac{r}{^{n} C _{r}}$ equals

(A)

(n - 1) a_{n}

(B)

n a_{n}

(C)

\frac{1}{2} n a_{n}

If $C_{r}$ stands for $^{n} C_{r}$ , then the sum of the series $\frac{2 ( \frac{n}{2} )! ( \frac{n}{2} )!}{n !} [C_{0}^{2} - 2 C_{1}^{2} + 3 C_{2}^{2} - .... + (- 1)^{n} (n + 1) C_{n}^{2}]$ , where $n$ is an even positive integer, is equal to

(A)

(B)

(- 1)^{n /2} (n + 1)

(C)

(- 1)^{n /2} (n + 2)

(D)

(- 1)^{n} n

JEE Advanced 1982

LEVELJEE Main

The value of the expression $^{47} C_{4} + \sum_{j = 1}^{5}^{52 - j} C_{3}$ is equal to

(A)

^{47} C_{5}

(B)

^{52} C_{5}

(C)

^{52} C_{4}

Let $m$ be the smallest positive integer such that the coefficient of $x^{2}$ in the expansion of $(1 + x)^{2} + (1 + x)^{3} + ... + (1 + x)^{49} + (1 + m x)^{50}$ is $(3 n + 1)^{51} C_{3}$ for some positive integer $n$ . Then the value of $n$ is .........

JEE Advanced 2013

LEVELBoard

The coefficients of three consecutive terms of $(1 + x)^{n + 5}$ are in the ratio $5 : 10 : 14$ . Then $n = .........$

JEE Advanced 1983

LEVELJEE Main

Let $n$ be positive integer. If the coefficients of 2nd, 3rd, and 4th terms in the expansion of $(1 + x)^{n}$ are in A.P., then the value of $n$ is .........

JEE Advanced 1985

LEVELJEE Main

Find the sum of the series : $\sum_{r = 0}^{n} (- 1)^{r}^{n} C_{r} [\frac{1}{2 ^{r}} + \frac{3 ^{r}}{2 ^{2 r}} + \frac{7 ^{r}}{2 ^{3 r}} + \frac{1 5 ^{r}}{2 ^{4 r}} \dots up to m terms]$ .

JEE Advanced 1982

LEVELBoard

The sum of the coefficients of the polynomial $(1 + x - 3 x^{2})^{2163}$ is .........

JEE Advanced 2003

LEVELJEE Main

Prove that $2^{k} (0 n) (k n) - 2^{k - 1} (1 n) (k - 1 n - 1) + 2^{k - 2} (2 n) (k - 2 n - 2) - .... + (- 1)^{k} (k n) (0 n - k) = (k n)$

JEE Advanced 2000

LEVELJEE Main

For any positive integer $m, n$ (with $n \geq m$ ), let $(m n) =^{n} C_{m}$ . Prove that $(m n) + (m n - 1) + (m n - 2) + .... + (m m) = (m + 1 n + 1)$ . Hence or otherwise, prove that $(m n) + 2 (m n - 1) + 3 (m n - 2) + .... + (n - m + 1) (m m) = (m + 2 n + 2)$ .

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The sum of coefficients of integral power of x in the binomial expansion (1−2x​)50 is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let S1​=∑j=110​j(j−1)10Cj​,S2​=∑j=110​j10Cj​ and S_3 = \sum_{j=1}^{10} j^2 ^{10}C_j. Statement-1: S3​=55×29. Statement-2: S1​=90×28 and S2​=10×28.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Statement-1: ∑r=0n​(r+1)nCr​=(n+2)2n−1. Statement-2: ∑r=0n​(r+1)nCr​xr=(1+x)n+nx(1+x)n−1.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The sum of the series 20C0​−20C1​+20C2​−20C3​+....+20C10​ is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The value of ^{50}C_4 + \sum_{r=1}^6 ^{56-r}C_3 is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the coefficients of rth, (r+1)th, and (r+2)th terms in the binomial expansion of (1+y)m are in A.P., then m and r satisfy the equation

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If Sn​=∑r=0n​nCr​1​ and tn​=∑r=0n​nCr​r​, then Sn​tn​​ is equal to

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If f(x)=xn, then the value of f(1)−1!f′(1)​+2!f′′(1)​−3!f′′′(1)​+⋯+n!(−1)nfn(1)​ is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The coefficients of xp and xq in the expansion of (1+x)p+q are

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the sum of the coefficients in the expansion of (a+b)n is 4096, then the greatest coefficient in the expansion is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } r and n are positive integers r>1,n>2 and coefficient of (r+2)th term and 3rth term in the expansion of (1+x)2n are equal, then n equals

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The value of (030​)(1030​)−(130​)(1130​)+(230​)(1230​)−....+(2030​)(3030​) is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The sum ∑i=0m​(i10​)(m−i20​), (where (qp​)=0 if p<q) is maximum when m is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } For 2≤r≤n, (rn​)+2(r−1n​)+(r−2n​)=

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If an​=∑r=0n​nCr​1​, then ∑r=0n​nCr​r​ equals

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The value of the expression 47C4​+∑j=15​52−jC3​ is equal to

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The coefficients of three consecutive terms of (1+x)n+5 are in the ratio 5:10:14. Then n=.........

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let n be positive integer. If the coefficients of 2nd, 3rd, and 4th terms in the expansion of (1+x)n are in A.P., then the value of n is .........

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Find the sum of the series : ∑r=0n​(−1)rnCr​[2r1​+22r3r​+23r7r​+24r15r​… up to m terms].

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The sum of the coefficients of the polynomial (1+x−3x2)2163 is .........

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Prove that 2k(0n​)(kn​)−2k−1(1n​)(k−1n−1​)+2k−2(2n​)(k−2n−2​)−....+(−1)k(kn​)(0n−k​)=(kn​)

The sum of coefficients of integral power of $x$ in the binomial expansion $(1 - 2 x)^{50}$ is

Let $S_{1} = \sum_{j = 1}^{10} j (j - 1)^{10} C_{j}, S_{2} = \sum_{j = 1}^{10} j^{10} C_{j}$ and $S_3 = \sum_{j=1}^{10} j^2 ^{10}C_j$ . Statement-1: $S_{3} = 55 \times 2^{9}$ . Statement-2: $S_{1} = 90 \times 2^{8}$ and $S_{2} = 10 \times 2^{8}$ .

Statement-1: $\sum_{r = 0}^{n} (r + 1)^{n} C_{r} = (n + 2) 2^{n - 1}$ . Statement-2: $\sum_{r = 0}^{n} (r + 1)^{n} C_{r} x^{r} = (1 + x)^{n} + n x (1 + x)^{n - 1}$ .

The sum of the series $^{20} C_{0} -^{20} C_{1} +^{20} C_{2} -^{20} C_{3} + .... +^{20} C_{10}$ is

The value of $^{50}C_4 + \sum_{r=1}^6 ^{56-r}C_3$ is

If the coefficients of $r$ th, $(r + 1)$ th, and $(r + 2)$ th terms in the binomial expansion of $(1 + y)^{m}$ are in A.P., then $m$ and $r$ satisfy the equation

If $S_{n} = \sum_{r = 0}^{n} \frac{1}{^{n} C _{r}}$ and $t_{n} = \sum_{r = 0}^{n} \frac{r}{^{n} C _{r}}$ , then $\frac{t _{n}}{S _{n}}$ is equal to

If $f (x) = x^{n}$ , then the value of $f (1) - \frac{f ^{'} ( 1 )}{1 !} + \frac{f ^{''} ( 1 )}{2 !} - \frac{f ^{'''} ( 1 )}{3 !} + \dots + \frac{( - 1 ) ^{n} f ^{n} ( 1 )}{n !}$ is

The coefficients of $x^{p}$ and $x^{q}$ in the expansion of $(1 + x)^{p + q}$ are

If the sum of the coefficients in the expansion of $(a + b)^{n}$ is 4096, then the greatest coefficient in the expansion is

$r$ and $n$ are positive integers $r > 1, n > 2$ and coefficient of $(r + 2)^{t h}$ term and $3 r^{t h}$ term in the expansion of $(1 + x)^{2 n}$ are equal, then $n$ equals

The value of $(0 30) (10 30) - (1 30) (11 30) + (2 30) (12 30) - .... + (20 30) (30 30)$ is

The sum $\sum_{i = 0}^{m} (i 10) (m - i 20)$ , (where $(q p) = 0$ if $p < q$ ) is maximum when $m$ is

For $2 \leq r \leq n$ , $(r n) + 2 (r - 1 n) + (r - 2 n) =$

If $a_{n} = \sum_{r = 0}^{n} \frac{1}{^{n} C _{r}}$ , then $\sum_{r = 0}^{n} \frac{r}{^{n} C _{r}}$ equals

The value of the expression $^{47} C_{4} + \sum_{j = 1}^{5}^{52 - j} C_{3}$ is equal to

The coefficients of three consecutive terms of $(1 + x)^{n + 5}$ are in the ratio $5 : 10 : 14$ . Then $n = .........$

Let $n$ be positive integer. If the coefficients of 2nd, 3rd, and 4th terms in the expansion of $(1 + x)^{n}$ are in A.P., then the value of $n$ is .........

Find the sum of the series : $\sum_{r = 0}^{n} (- 1)^{r}^{n} C_{r} [\frac{1}{2 ^{r}} + \frac{3 ^{r}}{2 ^{2 r}} + \frac{7 ^{r}}{2 ^{3 r}} + \frac{1 5 ^{r}}{2 ^{4 r}} \dots up to m terms]$ .

The sum of the coefficients of the polynomial $(1 + x - 3 x^{2})^{2163}$ is .........

Prove that $2^{k} (0 n) (k n) - 2^{k - 1} (1 n) (k - 1 n - 1) + 2^{k - 2} (2 n) (k - 2 n - 2) - .... + (- 1)^{k} (k n) (0 n - k) = (k n)$