$\mathop \sum \limits_{r = 0}^n \frac{{{{\left( { - 1} \right)}^r}}}{{{\;^n}{C_r}}}$ equals

$\frac{{2\left( {n + 1} \right)}}{{\left( {n + 2} \right)}}$

$\frac{n}{{n + 1}}\left[ {1 + {{\left( { - 1} \right)}^n}} \right]$

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JEE • MATHEMATICS • COMBINATIONS

Combinations practice · Easy

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Practice format and chapter quiz

These sets use single-correct-answer MCQs. They do not cover every JEE Main or JEE Advanced question format and are not full-length mock papers.

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Showing 1-30 of 236 questions

1 Easy

1 of 236

Application +4 | -1

Team A consists of 7 boys and n girls and Team B has 4 boys and 6 girls. If a total of 52 single matches can be arranged between these two teams, when a boy plays against a boy and a girl plays against a girl, then n is equal to

View answer and explanation

Correct option: C

Option C (4) is correct because it satisfies the equation formed by the total number of matches between boys and girls of the two teams.

2 Easy

2 of 236

Application +4 | -1

The number of ways choosing a committee of 4 woman and 5 men from 10 women and 9 men, if Mr.

A

refuses to serve on the committee when Ms.

B

is a member of the committee, is

View answer and explanation

Correct option: A

20580

Option A (20580) is correct because it is calculated by subtracting the number of committees with both Mr. A and Ms. B from the total number of possible committees.

3 Easy

3 of 236

Application +4 | -1

A man has 7 friends. In how many ways he can invite one or more of them for a tea party?

View answer and explanation

Correct option: C

127

Option C (127) is correct because each friend can either be invited or not, leading to 2⁷ possibilities, but we subtract 1 for the case where no one is invited.

4 Easy

4 of 236

Application +4 | -1

{\;^{n + 2}}{C_8}:{\;^{n - 2}}{P_4} = \frac{{57}}{{16}}

, then

n

is equal to

View answer and explanation

Correct option: A

Option A (19) is correct because solving the given equation involving combinations and permutations leads to n = 19.

5 Easy

5 of 236

Application +4 | -1

If the ratio

{ }^{2 n} {C}_3:{ }^n {C}_3

is equal to

11: 1, n

equals

View answer and explanation

Correct option: A

Option A (6) is correct because simplifying the ratio

{ }^{2 n} {C}_3:{ }^n {C}_3 = 11: 1

leads to the equation

8n - 4 = 11n - 22

, which gives

n=6

6 Easy

6 of 236

Application +4 | -1

25 lines are drawn in a plane such that no two of them are parallel and no three of them are concurrent. The number of points in which these lines intersect, is

View answer and explanation

Correct option: A

300

Option A (300) is correct because it represents the number of ways to choose 2 lines out of 25, which corresponds to the number of intersection points.

7 Easy

7 of 236

Application +4 | -1

The number of four letter words can be formed from the letters of the word INFINITY so that 3 are alike, one is different is

View answer and explanation

Correct option: C

Option C (16) is correct because it represents the number of four-letter words that can be formed from the letters of the word INFINITY such that 3 are alike and one is different.

8 Easy

8 of 236

Application +4 | -1

A father with 8 children takes 3 at a time to the Zoological Gardens, as often as he can without taking the same 3 children together more than once. The number of times each child will go to the garden is

View answer and explanation

Correct option: B

Option B (21) is correct because each child goes to the garden with every possible pair of the other 7 children, which is

{{\;^7}{C_2}}

9 Easy

9 of 236

Application +4 | -1

A lady gives a dinner party to 5 guests to be selected from nine friends. The number of ways of forming the party of 5, given that two of the friends will not attend the party together is

View answer and explanation

Correct option: C

Option C (91) is correct because it calculates the total number of ways to choose 5 guests from 9, then subtracts the number of ways where the two friends are both included.

10 Easy

10 of 236

Analyzing +4 | -1

{\;^n}{C_{r - 1}} = 36,{\;^n}{C_r} = 84

and

{\;^n}{C_{r + 1}} = 126,

then

View answer and explanation

Correct option: B

n = 9,r = 3

Option B (

n = 9, r = 3

) is correct because it satisfies the given combinatorial equations when the ratios of consecutive binomial coefficients are calculated.

11 Easy

11 of 236

Application +4 | -1

{ }^{2 {n}} C_4,{ }^{2 {n}} C_5

and

{ }^{2 {n}} C_6

are in A.P, then

n

is equal to

View answer and explanation

Correct option: C

Option C (7) is correct because solving the given equation involving combinations leads to a quadratic equation with n=7 as a valid solution.

12 Easy

12 of 236

Application +4 | -1

A polygon has 65 diagonals. The number of its sides is

View answer and explanation

Correct option: D

Option D (13) is correct because the number of sides of a polygon can be found using the formula for the number of diagonals.

13 Easy

13 of 236

Application +4 | -1

n \ge 2

is a positive integer, then the sum of the series

^{n + 1}{C_2} + 2{(^2}{C_2}{ + ^3}{C_2}{ + ^4}{C_2} + ...{ + ^n}{C_2})

View answer and explanation

Correct option: B

\frac{{n\,(n + 1)\,\,(2n + 1)}}{6}

Option B (

\frac{{n\,(n + 1)\,(2n + 1)}}{6}

) is the correct sum of the series, derived using combinatorial identities and simplification.

14 Easy

14 of 236

Application +4 | -1

The value of the expression

{ }^{47} C_4+\sum\limits_{{j=1}}^5 {}^{(52-j)} C_3

is equal to

View answer and explanation

Correct option: C

{ }^{52} C_4

Option C (

{ }^{52} C_4

) is correct because the expression simplifies to

{ }^{52} C_4

using the identity

{ }^{n+1} C_r={ }^n C_r+{ }^n C_{r-1}

repeatedly.

15 Easy

15 of 236

Application +4 | -1

\sum \limits_{r = 0}^n \frac{{{{\left( { - 1} \right)}^r}}}{{{\;^n}{C_r}}}

equals

View answer and explanation

Correct option: A

\frac{{\left( {n + 1} \right)}}{{\left( {n + 2} \right)}}\left[ {1 + {{\left( { - 1} \right)}^n}} \right]

Option A is correct because the given summation can be simplified using properties of binomial coefficients and telescoping series.

Explore Other Chapters in MATHEMATICS

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SEQUENCE OF SERIES 1764 MCQs DEFINITE INTEGRATION 1443 MCQs MATRICES 1363 MCQs INDEFINITE INTEGRATION 1179 MCQs PROBABILITY 1146 MCQs COMPLEX NUMBERS 980 MCQs

16 Easy

16 of 236

Application +4 | -1

Kunal Gaba has

n

objects, each of weight

w

. He weighs them in pairs and finds the sum of the weights of all possible pairs is

120 \mathrm{~g}

. When his friend Rakshit weighs them in triplets, the sum of all possible weights is

240 \mathrm{~g}

. The value of

n

View answer and explanation

Correct option: B

Option C (4032) is correct because it is calculated by subtracting the number of selections with three consecutive numbers from the total number of selections.

17 Easy

17 of 236

Application +4 | -1

a,b,c,d,e

are prime integers, then the number of divisors of

a{b^2}{c^2}de

excluding 1 as a factor is

View answer and explanation

Correct option: D

Option D (71) is correct because the number of divisors is calculated by adding 1 to each exponent in the prime factorization, multiplying the results, and then subtracting 1 to exclude 1 as a factor.

18 Easy

18 of 236

Application +4 | -1

Out of 8 given points, 3 are collinear. How many different straight lines can be drawn by joining any two points from those 8 points?

View answer and explanation

Correct option: A

Option A (26) is correct because it calculates the number of lines formed by 8 points, accounting for the collinearity of 3 points.

19 Easy

19 of 236

Application +4 | -1

A polygon has 54 diagonals. Number of sides of this polygon is

View answer and explanation

Correct option: A

Option A (12) is correct because the number of sides of the polygon is found by solving the quadratic equation derived from the formula for the number of diagonals.

20 Easy

20 of 236

Understanding +4 | -1

n

is even and

{ }^n C_0 < { }^n C_1 < { }^n C_2 < \cdots < n C_r>{ }^n C_{r+1}>\cdots > { }^n C_n

then

r=

View answer and explanation

Correct option: A

\frac{n}{2}

Option A (

\frac{{n}}{{2}}

) is correct because for even n, the binomial coefficient

{ }^n C_r

is maximum when r = n/2.

21 Easy

21 of 236

Application +4 | -1

The total number of all proper factors of 75600 is

View answer and explanation

Correct option: C

118

Option C (118) is correct because it calculates the total number of factors and subtracts 1 (for the number itself) to get the number of proper factors.

22 Easy

22 of 236

Application +4 | -1

{\;^n}{C_{n - r}} + 3.{\;^n}{C_{n - r + 1}} + 3.{\;^n}{C_{n - r + 2}} + {\;^n}{C_{n - r + 3}} = {\;^x}{C_r},

then

x =

View answer and explanation

Correct option: C

n + 3

Option C (

n + 3

) is correct because simplifying the given expression using the combination identity leads to

{\;^{n + 3}}{C_r} = {\;^x}{C_r}

, which implies

x = n + 3

23 Easy

23 of 236

Application +4 | -1

At an election, three wards of a town are can vassesed by 3,4 and 5 men respectively. If 20 men volunteer, then the number ways can they be allotted to the different wards is

View answer and explanation

Correct option: D

{ }^{20} C_3 \cdot{ }^{17} C_4 \cdot{ }^{13} C_5

Option D (

{}^{20} C_3 \cdot{}^{17} C_4 \cdot{}^{13} C_5

) is correct because it represents the number of ways to choose 3 men for the first ward, then 4 from the remaining for the second, and 5 from the rest for the third.

24 Easy

24 of 236

Analyzing +4 | -1

The number of ways of selecting 4 letters from 3 'a's, 4 'b's and 5 'c's is

View answer and explanation

Correct option: A

\frac{{ }^8 \mathrm{C}_2}{2}

Correct Answer:

\frac{{ }^8 \mathrm{C}_2}{2}

is incorrect. The correct answer is 77. The number of ways to select 4 letters is found by considering all possible combinations of a's, b's, and c's, taking into account the available quantities of each letter.

25 Easy

25 of 236

Application +4 | -1

The maximum number of points into which 4 circles and 4 straight lines intersect is

View answer and explanation

Correct option: B

The correct answer is B (50) because it represents the maximum number of intersection points between 4 circles and 4 lines.

26 Easy

26 of 236

Application +4 | -1

The number of parallelograms that can be formed from a set of four parallel lines intersecting another set of three parallel lines, is

View answer and explanation

Correct option: C

Option C (12) is correct because it represents the number of ways to choose two lines from each set of parallel lines to form a parallelogram.

27 Easy

27 of 236

Application +4 | -1

Everybody in a room shakes hands with everybody else. The total number of hand shakes is 66. The total number of persons in the room is

View answer and explanation

Correct option: B

Option B (12) is correct because the number of handshakes is given by

{{\;^n}{C_2}} = 66

, which solves to

n = 12

28 Easy

28 of 236

Application +4 | -1

A car will hold 2 in the front seat and 1 in the rear seat. If among 6 persons 2 can drive, then number of ways in which the car can be filled, is

View answer and explanation

Correct option: B

Option B (20) is correct because it accounts for selecting a driver and then selecting the remaining passengers.

29 Easy

29 of 236

Analyzing +4 | -1

A bag contains 3 black, 4 white and 2 red balls, all the balls being different. The number of selections of at most 6 balls containing balls of all the colours, is

View answer and explanation

Correct option: A

42\left( {4!} \right)

Option A (

42(4!)

) is correct because it is found by considering all possible combinations of selecting at most 6 balls with at least one of each color.

30 Easy

30 of 236

Application +4 | -1

{ }^{{n}} P_{{r}}=1680

and

{ }^{{n}} C_{{r}}=70

, then

n

is equal to

View answer and explanation

Correct option: C

Option C (8) is correct because using the relationship between permutations and combinations, we find r=4, and then solving the permutation equation leads to n=8.

Revision notes and key takeaways

Review COMBINATIONS in your textbook or class notes, then use this set to check your understanding.

Key Takeaways

✓ This set contains 236 single-correct-answer MCQs on COMBINATIONS.
✓ Try each question before opening its answer. Read the explanation where one is available.
✓ Sign in to save submitted answers. Revisit incorrect answers alongside your textbook or class notes.

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