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What will be the remainder when $(67^{67} + 67)$ is divided by $68$ ?

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Important Questions on Binomial Theorem

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

EASY

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

The remainder when

2^{2000}

is divided by

17

EASY

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

{(27)}^{999}

is divided by

7

, then the remainder is

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

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})

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

HARD

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

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

EASY

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

The sum of the coefficients in the expansion of

{(1 + 2 x - x^{2})}^{20}

MEDIUM

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

If the sum of the coefficients in the expansions of

{(a^{2} x^{2} - 2 a x + 1)}^{51}

is zero, then

a

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 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:

HARD

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

The fractional part of a real number

x

x - [x],

where

[x]

is the greatest integer less than or equal to

x .

Let

F_{1}

and

F_{2}

be the fractional parts of

{(44 - \sqrt{2017})}^{2017}

and

{(44 + \sqrt{2017})}^{2017}

respectively. Then

F_{1} + F_{2}

lies between the numbers

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

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

MEDIUM

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

{}^{11}C_{0} + {}^{11}C_{1} + {}^{11}C_{2} + {}^{11}C_{3} + {}^{11}C_{4} + {}^{11}C_{5} =

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

HARD

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

{p}

denotes the fractional part of the number

p,

then

\{\frac{3^{200}}{8}\}

is equal to

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

MEDIUM

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

The digit in the unit place of

7^{291}

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

What will be the remainder when 6767+67 is divided by 68?

Important Questions on Binomial Theorem

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What will be the remainder when $(67^{67} + 67)$ is divided by $68$ ?