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If $a, b, c, d$ are odd natural numbers such that $a + b + c + d = 20$ then the number of values of $a, b, c, d,$ is

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Important Questions on Binomial Theorem

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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,

MEDIUM

Mathematics>Algebra>Binomial Theorem>Multinomial Theorem

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!}

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

The value of

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

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

The total number of irrational terms in the binomial expansion of

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

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}

HARD

Mathematics>Algebra>Binomial Theorem>Multinomial Theorem

Let

X

be a set containing

10

elements and

P (X)

be its power set. If

A

and

B

are picked up at random from

P (X)

, with replacement, then the probability that

A

and

B

have equal number of elements is:

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 ______

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

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 ______ .

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-

HARD

Mathematics>Algebra>Binomial Theorem>Multinomial Theorem

The term independent of $x$ in the expansion of

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

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

MEDIUM

Mathematics>Algebra>Binomial Theorem>Multinomial Theorem

The term independent of

x

in the binomial expansion of

(1 - \frac{1}{x} + 3 x^{5}) {(2 x^{2} - \frac{1}{x})}^{8}

MEDIUM

Mathematics>Algebra>Binomial Theorem>Multinomial Theorem

If the coefficient of the three successive terms in the binomial expansion of

{(1 + x)}^{n}

are in the ratio

1 : 7 : 42

, then the first of these terms in the expansion is

HARD

Mathematics>Algebra>Binomial Theorem>Multinomial Theorem

If in a regular polygon the number of diagonals is

54

, then the number of sides of this polygon is:

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

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

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}\}

If a, b, c,d are odd natural numbers such that a+b+c+d=20 then the number of values of a, b, c, d, is

Important Questions on Binomial Theorem

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If $a, b, c, d$ are odd natural numbers such that $a + b + c + d = 20$ then the number of values of $a, b, c, d,$ is