EASY

Earn 100

Compute ${(98)}^{5}$ using binomial theorem.

Important Questions on Binomial Theorem

EASY

Mathematics>Algebra>Binomial Theorem>Binomial Theorem for Positive Integral Index

The number of terms in the expansion of

{(x^{2} + y^{2})}^{25} - {(x^{2} - y^{2})}^{25}

after simplification is

HARD

Mathematics>Algebra>Binomial Theorem>Binomial Theorem for Positive Integral Index

The number of terms in the expansion of ${(1 + x)}^{101} {(1 - x + x^{2})}^{100}$ in powers of $x$ is

MEDIUM

Mathematics>Algebra>Binomial Theorem>Binomial Theorem for Positive Integral Index

If some three consecutive coefficients in the binomial expansion of

{(x + 1)}^{n}

in powers of

x

are in the ratio

2 : 15 : 70,

then the average of these three coefficients is:

EASY

Mathematics>Algebra>Binomial Theorem>Binomial Theorem for Positive Integral Index

If the sum of the coefficients in the expansion of

{(x + y)}^{n}

1024,

then the value of the greatest coefficient in the expansion, is

EASY

Mathematics>Algebra>Binomial Theorem>Binomial Theorem for Positive Integral Index

\frac{1}{9!} + \frac{1}{3! 7!} + \frac{1}{5! 5!} + \frac{1}{7! 3!} + \frac{1}{9!}

Is equal to

EASY

Mathematics>Algebra>Binomial Theorem>Binomial Theorem for Positive Integral Index

[x]

represents the greatest integer not greater than

x,

then

[{(1 + \frac{1}{10000})}^{10000}] =

MEDIUM

Mathematics>Algebra>Binomial Theorem>Binomial Theorem for Positive Integral Index

The coefficient of

x^{1012}

in the expansion of

{(1 + x^{n} + x^{253})}^{10},

(where

n \leq22

is any positive integer), is

HARD

Mathematics>Algebra>Binomial Theorem>Binomial Theorem for Positive Integral Index

1 \times n

rectangle

(n \geq 1)

is divided into

n

unit

(1 \times 1)

squares. Each square of this rectangle is coloured red, blue or green. Let

f (n)

be the number of colourings of the rectangle in which there are an even number of red squares. What is the largest prime factor of

\frac{f (9)}{f (3)}

? (The number of red squares can be zero.)

HARD

Mathematics>Algebra>Binomial Theorem>Binomial Theorem for Positive Integral Index

{(\sqrt{3} + \sqrt{2})}^{4} - {(\sqrt{3} - \sqrt{2})}^{4} =

HARD

Mathematics>Algebra>Binomial Theorem>Binomial Theorem for Positive Integral Index

If the coefficients of

x^{3}

and

x^{4}

in the expansion of

(1 + a x + b x^{2}) {(1 - 2 x)}^{18}

in powers of

x

are both zero, then

(a, b)

is equal to

MEDIUM

Mathematics>Algebra>Binomial Theorem>Binomial Theorem for Positive Integral Index

The expression

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

is a polynomial in

x

of degree

MEDIUM

Mathematics>Algebra>Binomial Theorem>Binomial Theorem for Positive Integral Index

The value of

(^{21} C_{1} -^{10} C_{1}) + (^{21} C_{2} -^{10} C_{2}) + (^{21} C_{3} -^{10} C_{3}) + (^{21} C_{4} -^{10} C_{4}) + \dots + (^{21} C_{10} -^{10} C_{10})

MEDIUM

Mathematics>Algebra>Binomial Theorem>Binomial Theorem for Positive Integral Index

Let

n

be a positive integer. Let

A = \sum_{k = 0}^{n} {(- 1)}^{k} \times {}^{n}C_{k} [{(\frac{1}{2})}^{k} + {(\frac{3}{4})}^{k} + {(\frac{7}{8})}^{k} + {(\frac{15}{16})}^{k} + {(\frac{31}{32})}^{k}]

. If

63 A = 1 - \frac{1}{2^{30}},

then

n

is equal to ______ .

MEDIUM

Mathematics>Algebra>Binomial Theorem>Binomial Theorem for Positive Integral Index

α

and

β

are the coefficients of

x^{8}

and

x^{- 24}

, respectively, in the expansion of

{(x^{4} + 2 + \frac{1}{x^{4}})}^{10}

in powers of

x,

then

\frac{α}{β}

is equal to

EASY

Mathematics>Algebra>Binomial Theorem>Binomial Theorem for Positive Integral Index

The number of dissimilar terms in the expansion of

{(a + b)}^{n}

n + 1

. So the number of dissimilar terms in the expansion of

{(a + b + c)}^{12}

HARD

Mathematics>Algebra>Binomial Theorem>Binomial Theorem for Positive Integral Index

A possible value of

x,

for which the ninth term in the expansion of

{\{3^{\log_{3} \sqrt{25^{x - 1} + 7}} + 3^{(- \frac{1}{8}) \log_{3} (5^{x - 1} + 1)}\}}^{10}

in the increasing powers of

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

is equal to

180,

is :

MEDIUM

Mathematics>Algebra>Binomial Theorem>Binomial Theorem for Positive Integral Index

The coefficient of

x^{3}

in the expansion of

{(1 - x + x^{2})}^{5}

MEDIUM

Mathematics>Algebra>Binomial Theorem>Binomial Theorem for Positive Integral Index

X = \{4^{n} - 3 n - 1 : n \in N\}

and

Y = \{9 (n - 1) : n \in N\}

, where

N

is the set of natural numbers, then

X \cup Y

is equal to

EASY

Mathematics>Algebra>Binomial Theorem>Binomial Theorem for Positive Integral Index

({}^{7}{C_{0}} + {}^{7}{C_{1}}) + ({}^{7}{C_{2} + {}^{7}{C_{3}}}) + \dots + ({}^{7}{C_{6} + {}^{7}{C_{7}}}) =

MEDIUM

Mathematics>Algebra>Binomial Theorem>Binomial Theorem for Positive Integral Index

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

Compute 985 using binomial theorem.

Important Questions on Binomial Theorem

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Compute ${(98)}^{5}$ using binomial theorem.