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The numbers of terms in the expansion of ${(a^{3} + \frac{1}{a^{3}} + 1)}^{100}$ is

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Important Questions on Binomial Theorem

HARD

Mathematics>Algebra>Binomial Theorem>Multinomial Theorem

The sum of coefficients of integral powers of

x

in the binomial expansion of

{(1 - 2 \sqrt{x})}^{50}

MEDIUM

Mathematics>Algebra>Binomial Theorem>Multinomial Theorem

Let

A = [a_{\hat{i} \hat{j}}] \forall a_{i j} \in Z \cap [0, 4], 1 \leq i, j \leq 2

. The number of matrices

A

such that the sum of all entries is

a

prime number

p \in (2, 13)

is _____ .

EASY

Mathematics>Algebra>Binomial Theorem>Multinomial Theorem

The coefficient of

x^{7}

{(1 - x + 2 x^{3})}^{10}

is __________ .

EASY

Mathematics>Algebra>Binomial Theorem>Multinomial Theorem

A ratio of the

5^{t h}

term from the beginning to the

5^{t h}

term from the end in the binomial expansion of

{(2^{\frac{1}{3}} + \frac{1}{2 {(3)}^{\frac{1}{3}}})}^{10}

HARD

Mathematics>Algebra>Binomial Theorem>Multinomial Theorem

Let

{(2 x^{2} + 3 x + 4)}^{10} = \sum_{r = 0}^{20} a_{r} x^{r} .

Then

\frac{a_{7}}{a_{13}}

is equal to ______

HARD

Mathematics>Algebra>Binomial Theorem>Multinomial Theorem

{\sum_{i = 1}^{20} (\frac{^{20} C_{i - 1}}{^{20} C_{i} +^{20} C_{i - 1}})}^{3} = \frac{k}{21},

then

k

equals

MEDIUM

Mathematics>Algebra>Binomial Theorem>Multinomial Theorem

The number of ways of giving $20$ distinct oranges to $3$ children such that each child gets at least one orange is $_____$

HARD

Mathematics>Algebra>Binomial Theorem>Multinomial Theorem

The total number of irrational terms in the binomial expansion of

{(7^{\frac{1}{5}} - 3^{\frac{1}{10}})}^{60}

EASY

Mathematics>Algebra>Binomial Theorem>Multinomial Theorem

The number of terms in the expansion of

{(x + y + z)}^{10}

HARD

Mathematics>Algebra>Binomial Theorem>Multinomial Theorem

If the constant term in the expansion of

{(3 x^{3} - 2 x^{2} + \frac{5}{x^{5}})}^{10}

2^{k} . l

, where

l

is an odd integer, then the value of

k

is equal to

HARD

Mathematics>Algebra>Binomial Theorem>Multinomial Theorem

The coefficient of

x^{2}

in the expansion of the product

(2 - x^{2}) \{{(1 + 2 x + 3 x^{2})}^{6} + {(1 - 4 x^{2})}^{6}\}

MEDIUM

Mathematics>Algebra>Binomial Theorem>Multinomial Theorem

If the number of terms in the expansion of

{(1 - \frac{2}{x} + \frac{4}{y^{2}})}^{n}, x, y \neq 0,

28

, then the sum of the coefficients of all the terms in this expansion is

MEDIUM

Mathematics>Algebra>Binomial Theorem>Multinomial Theorem

The positive value of

λ

for which the co-efficient of

x^{2}

in the expansion

x^{2} {(\sqrt{x} + \frac{λ}{x^{2}})}^{10}

720,

HARD

Mathematics>Algebra>Binomial Theorem>Multinomial Theorem

If the third term in the binomial expansion of

{(1 + x^{\log_{2} x})}^{5}

equals

2560,

then a possible value of

x

MEDIUM

Mathematics>Algebra>Binomial Theorem>Multinomial Theorem

\sum_{r = 0}^{25} \{({}^{50}C_{r}) ({}^{50 - r}C_{25 - r})\} = K ({}^{50}C_{25})

, then

K

is equal to

HARD

Mathematics>Algebra>Binomial Theorem>Multinomial Theorem

Let

X = {(^{10} C_{1})}^{2} + 2 {(^{10} C_{2})}^{2} + 3 {(^{10} C_{3})}^{2} + \dots + 10 {(^{10} C_{10})}^{2}

, where

^{10} C_{r}, r \in \{1,2, \dots ., 10\}

denote binomial coefficients. Then, the value of

\frac{1}{1430} X

is ______ .

MEDIUM

Mathematics>Algebra>Binomial Theorem>Multinomial Theorem

The value of

[C (7, 0) + C (7, 1)] + [C (7, 1) + C (7, 2)] + \dots + [C (7, 6) + C (7, 7)]

MEDIUM

Mathematics>Algebra>Binomial Theorem>Multinomial Theorem

The constant term in the expansion of

{(2 x + \frac{1}{x^{7}} + 3 x^{2})}^{5}

is _____ .

EASY

Mathematics>Algebra>Binomial Theorem>Multinomial Theorem

The coefficient of

x^{4}

in the expansion of

{(1 + x + x^{2})}^{10}

is ________

HARD

Mathematics>Algebra>Binomial Theorem>Multinomial Theorem

Let ${(1 + x + x^{2})}^{2014} = a_{0} + a_{1} x + a_{2} x^{2} + a_{3} x^{3} + \dots + a_{4028} x^{4028},$ and let

$A = a_{0} - a_{3} + a_{6} - \dots + a_{4026},$

$B = a_{1} - a_{4} + a_{7} - \dots - a_{4027},$

$C = a_{2} - a_{5} + a_{8} - \dots + a_{4028},$

Then-

The numbers of terms in the expansion of a3+1a3+1100 is

Important Questions on Binomial Theorem

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The numbers of terms in the expansion of ${(a^{3} + \frac{1}{a^{3}} + 1)}^{100}$ is