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The number of integers between $100$ to $1000$ whose sum of digits is $14$

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Important Questions on Binomial Theorem

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

EASY

Mathematics>Algebra>Binomial Theorem>Multinomial Theorem

The coefficient of

x^{7}

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

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}\}

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 _____ .

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 ______

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 _____ .

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 $_____$

EASY

Mathematics>Algebra>Binomial Theorem>Multinomial Theorem

The coefficient of

x^{4}

in the expansion of

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

is ________

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 number of ordered triplets of positives integers which are solutions of the equation

x + y + z = 100

, is

MEDIUM

Mathematics>Algebra>Binomial Theorem>Multinomial Theorem

Number of ways in which 25 identical balls can be distributed among Ram, Shyam, Sunder and Ghanshyam such that atleast 1, 2, 3, and 4 balls are given to Ram, Shyam, Sunder and Ghanshyam respectively, is-

MEDIUM

Mathematics>Algebra>Binomial Theorem>Multinomial Theorem

Coefficient of

t^{24}

{(1 + t^{2})}^{12} (1 + t^{12}) (1 + t^{24})

HARD

Mathematics>Algebra>Binomial Theorem>Multinomial Theorem

Number of ways in which

15

indistinguishable oranges can be put in

3

different boxes so that every box has at most

8

oranges

MEDIUM

Mathematics>Algebra>Binomial Theorem>Multinomial Theorem

In how many ways can Rs 16 be divided into 4 persons when none of them get less than Rs 3?

MEDIUM

Mathematics>Algebra>Binomial Theorem>Multinomial Theorem

Coefficient of

x^{2} y^{3} z^{4}

{(a x + b y + c z)}^{9}

MEDIUM

Mathematics>Algebra>Binomial Theorem>Multinomial Theorem

If in the expansion of

{(1 + x)}^{m} {(1 - x)}^{n},

the coefficient of

x

and

x^{2}

are

3

and

- 6

respectively, then

m

EASY

Mathematics>Algebra>Binomial Theorem>Multinomial Theorem

The sum of the coefficient in

{(x + 2 y + z)}^{10}

EASY

Mathematics>Algebra>Binomial Theorem>Multinomial Theorem

The number of terms in the expansion of

{(a + b + c)}^{10}

HARD

Mathematics>Algebra>Binomial Theorem>Multinomial Theorem

a, b, c

are positive integers such that

a + b + c \leq 8

, then the number of possible values of the ordered triplets

(a, b, c)

MEDIUM

Mathematics>Algebra>Binomial Theorem>Multinomial Theorem

a, b, c

be three natural numbers in A.P. and

a + b + c = 21

, then the possible number of values of

a, b, c

is-

The number of integers between 100 to 1000 whose sum of digits is 14

Important Questions on Binomial Theorem

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The number of integers between $100$ to $1000$ whose sum of digits is $14$