Permutations & Combinations Questions - JEE

Find the number of diagonals in a quindecagon (15-sided polygon).

Number of cyphers (trailing zeros) at the end of $\binom{2016}{1008}$.

A student is to answer 10 out of 13 questions in an examination such that he must choose at least 4 from the first five questions. The number of choices available to him is

Number of 5-digit natural numbers such that product of their digits equals number of ways to create a necklace out of 6 beads from 10 distinct beads is

Sum of all even divisors of the number $N = 2016$.

Let n men participated in a tournament. The number of matches men played with 2 women a and b is 2(n + n) = 4n. Number of matches men played among themselves is 2nC2. According to the question, 2nC2 = 4n + 66. The number of men which lie in the interval [10, 12] is:

If ${}^nC_r$ denotes the number of combinations of $n$ things taken $r$ at a time, then the expression ${}^nC_{r+1} + {}^nC_{r-1} + 2 \times {}^nC_r$ equals

In how many ways can 5 identical black balls, 7 identical red balls and 6 identical green balls be arranged in a row so that at least one ball is separated from balls of the same colour?

In a shop there are five types of ice-creams available. A child buys six ice-creams.Statement-1: The number of different ways the child can buy the six ice-creams is ${}^{10}C_5$.Statement-2: The number of different ways the child can buy the six ice-creams is equal to the number of different ways of arranging 6 A's and 4 B's in a row.

If in a regular polygon the number of diagonals is 54, then the number of sides of this polygon is

If $\dfrac{{}^{n+2}C_6}{{}^{n-2}P_2} = 11$, then $n$ satisfies the equation:

Fifteen coupons are numbered 1, 2, 3, . . . 15. In how many ways seven coupons are selected such that the largest number appearing on the selected coupon is 9?

From 6 different novels and 3 different dictionaries, 4 novels and 1 dictionary are to be selected and arranged in a row on a shelf so that the dictionary is always in the middle. The number of such arrangements is

How many six-digit odd numbers, greater than 6,00,000, can be formed from the digits 5, 6, 7, 8, 9, and 0 if(a) repetition of digits is allowed?

The number of integers greater than 6,000 that can be formed, using the digits 3, 5, 6, 7 and 8, without repetition, is

If $\dfrac{^{n+2}C_6}{^{n-2}P_2} = 11$, then the value of $n$ is:

Number of 7 letter smart words is

The total number of functions, f:{1, 2, 3, 4} $\rightarrow$ {1, 2, 3, 4, 5, 6} such that f(1) + f(2) = f(3), is equal to:

Let $A=\sum_{i=0}^{2023}\sum_{j=0}^{2023}\min\{\binom{2023}{i},\binom{2023}{j}\}$ and $B=\sum_{i=0}^{2023}\sum_{j=0}^{2023}\max\{\binom{2023}{i},\binom{2023}{j}\}$. Then $A+B$ equals

Find the number of paths from (0, 0) to (n, n) such that y \leq x at every lattice point on the path.

If in a regular polygon the number of diagonals is 54, then the number of sides of this polygon is

How many numbers greater than hundred and divisible by 5 can be made from the digits 3, 4, 5, 6, if no digit is repeated?

Four-digit numbers are formed using digits from $\{0,1,2,3,4,5\}$, repetition allowed. Statement $S_1$: the number of such odd numbers is 480. Statement $S_2$: the number formed with exactly three different digits is 360.

With normal method, find the highest power of 5 in 137!.

How many numbers can be formed from the digits 1, 2, 2, 5, 6 which are greater than 50000?

Ten IIT and 2 DCE students sit in a row. The number of ways in which exactly 3 IIT students sit between 2 DCE students is

Find the total number of positive integral solutions for $(x, y, z)$ such that $xyz = 24$. Also find the total number of integral solutions.

In an election, the number of candidates is one greater than the persons to be elected. If a voter can vote in 254 ways, the number of candidates is

Messages are conveyed by arranging four white, one blue, and three red flags on a pole. Flags of the same color are alike. If a message is transmitted by the order in which the colours are arranged, the total number of messages that can be transmitted if exactly six flags are used is

From six different novels and three different dictionaries, four novels and one dictionary are to be selected and arranged in a row on the shelf so that the dictionary is always in the middle. Then the number of such arrangements is

From 6 different novels and 3 different dictionaries, 4 novels and 1 dictionary are to be selected and arranged in a row on a shelf so that the dictionary is always in the middle. The number of such arrangements is

Assuming the balls to be identical except for difference in colours, the number of ways in which one or more balls can be selected from 10 white, 9 green and 7 black balls is

Eleven animals of a circus have to be placed in eleven cages (one in each cage). If 4 of the cages are too small for 6 of the animals, find the number of the ways of caging all the animals.

The total number not more than 20 digits that are formed by using the digits 0, 1, 2, 3, and 4 is

There are six periods in each working day of a school. Find the number of ways in which 5 subjects can be arranged if each subject is allotted at least one period and no period remains vacant.

If $a, b, c, d$ are odd natural numbers such that $a + b + c + d = 20$, then find the number of values of the ordered quadruplet $(a, b, c, d)$.

In a polygon, no three diagonals are concurrent. If the total number of points of intersection of diagonals interior to the polygon is 70, then the number of diagonals of the polygon is

Find the value(s) of r satisfying the equation ${}^{69}C_{3r-1} - {}^{69}C_{r^2} = {}^{69}C_{r^2-1} - {}^{69}C_{3r}$.

The number of solutions $(n_1, n_2)$ which satisfy $n_1 n_2 = 2n_1 - n_2$, where $n_1, n_2 \in \mathbb{I}$ (integers) is

A train timetable must be compiled for various days of the week so that two trains twice a day depart for three days, one train daily for two days, and three trains once a day for two days. How many different timetables can be compiled?

The sum of all the numbers of four different digits that can be made by using the digits 0, 1, 2, and 3 is

The number of five-digit numbers that contain 7 exactly once is

In a city no two persons have identical set of teeth and there is no person without a tooth. Also no person has more than 32 teeth. If we disregard the shape and size of tooth and consider only the positioning of the teeth, the maximum population of the city is

A candidate is required to answer 6 out of 10 questions, which are divided into two groups, each containing 5 questions. He is not permitted to attempt more than 4 questions from either group. The number of different ways in which the candidate can choose 6 questions is

The streets of a city are arranged like the lines of a chessboard. There are m streets running from north to south and n streets from east to west. Find the number of ways in which a man can travel from north-west to south-east corner, covering the shortest possible distance.

In how many ways five persons can stand in a row?

Words of length 10 are formed using the letters $A, B, C, D, E, F, G, H, I, J$. Let $x$ be the number of such words where no letter is repeated; and let $y$ be the number of such words where exactly one letter is repeated twice and no other letter is repeated. Then $\dfrac{y}{9x}$ = ________.

Find the total number of six-digit numbers in which all and only odd digits appear.

If N denotes the number of ways of selecting r objects out of n distinct objects (r ≥ n) with unlimited repetition but with each object included at least once in selection, then N is equal to

Let $A = \{x_1, x_2, x_3, \ldots, x_7\}$, $B = \{y_1, y_2, y_3\}$. The total number of functions $f: A \to B$ that are on to and there are exactly three elements $x$ in $A$ such that $f(x) = y_2$ is equal to

Permutations & Combinations Questions (855)