Permutations & Combinations Questions (855)

The number of ways to fill each of the four cells of the table with a distinct natural number such that the sum of the numbers is 10 and the sums of the numbers placed diagonally are equal is

Find the number of ways in which letters A, A, A, B, B, B can be placed in the squares of the figure (a 3×2 grid) so that no row remains empty.

Find the number of ways of selecting 3 pairs from 8 distinct objects.

X has 7 friends, 3 of them are males (M) and 4 are females (L). Y has 7 friends, 4 of them are males (M) and 3 are females (L). A party of 3 from X's friends and 3 from Y's friends is to be arranged such that there are 3 males and 3 females in the party. The total number of ways of arranging the party is:

At least 4 from the first 5 questions means either 4 or 5 from the first 5 questions, and the rest from the remaining 8 questions. The number of ways to attempt a paper of 13 questions (at least 4 from first 5) choosing exactly 6 questions in total is:

Total number of six-digit numbers that can be formed having the property that every succeeding digit is greater than the preceding digit is equal to

There are \(n\) distinct white and \(n\) distinct black balls. If the number of ways of arranging them in a row so that neighboring balls are of different colors is 1152, then the value of \(n\) is ___.

In an experiment, \(n\) six-faced normal dice are thrown. Find the number of sets of observations which are indistinguishable among themselves.

If \(T_n\) = Number of triangles formed using the vertices of an \(n\)-sided regular polygon and \(T_{n+1} - T_n = 21\), then \(n\) equals:

A person has \(n\) friends. The minimum value of \(n\) so that a person can invite a different pair of friends every day for four weeks in a row is ___.

In how many different ways can a set A of 3n elements be partitioned into 3 subsets of equal number of elements? (The subsets P, Q, R form a partition if \(P \cup Q \cup R = A,\ P \cap R = \phi,\ Q \cap R = \phi,\ R \cap P = \phi\).)

Find the coefficient of \(x^{98}\) in the continued product \((x+1)(x+2)(x+3)\cdots(x+100)\).

In a conference 10 speakers are present. If \(S_1\) wants to speak before \(S_2\) and \(S_2\) wants to speak after \(S_3\), then find the number of ways all the 10 speakers can give their speeches with the above restriction if the remaining seven speakers have no objection to speak at any number.

In a room, there are 12 bulbs of the same wattage, each having a separate switch. The number of ways to light the room with different amount of illumination is

There were two women participating in a chess tournament. Every participant played two games with the other participates. The number of games that the men played among themselves proved to exceed by 66 number of games that the men played with the women. The number of participants is

How many even numbers of four digits can be made with the digits 0, 3, 5, 4? Find the sum of the numbers.

8-digit numbers are formed using the digits 1, 1, 2, 2, 2, 3, 4, 4. The number of such numbers in which the odd digits do not occupy odd places, is

In a three-storey building, there are four rooms on the ground floor, two on the first and two on the second floor. If the rooms are to be allotted to six persons, one person occupying one room only, the number of ways in which this can be done so that no floor remains empty is

Number of ways in which ₹18 can be distributed among four persons such that nobody receives less than ₹3 is

Consider the letters of the word MATHEMATICS. Possible number of words taking all letters at a time such that at least one repeating letter is at odd position in each word is

How many automobile license plates can be made, if each plate contains two different letters followed by three different digits?

In the decimal system of numeration of six-digit numbers in which the sum of the digits is divisible by 5 is

Let n and k be positive integers such that \( n \geq \dfrac{k(k+1)}{2} \). Find the number of solutions \((x_1, x_2, \ldots, x_k)\), \(x_1 \geq 1,\, x_2 \geq 2,\, \ldots,\, x_k \geq k\), all integers satisfying the condition \(x_1 + x_2 + x_3 + \cdots + x_k = n\).

There are six teachers. Out of them two are primary teachers, two are middle teachers, and two are secondary teachers. They are to stand in a row, so as the primary teachers, middle teachers, and secondary teachers are always in a set. Find the number of ways in which they can do so.

In a group of 13 cricket players, 4 are bowlers. Then in how many ways can they form a cricket team of 11 players in which at least 2 bowlers are included?

In a particular programming language, a valid variable name can consist of a sequence of one to six alphanumeric characters A, B, C, ..., Z, 0, 1, 2, ..., 9 beginning with a letter. Find the total number of valid variable names.

The number of ways to give 16 different things to three persons \(A\), \(B\), \(C\) so that \(B\) gets one more than \(A\) and \(C\) gets two more than \(B\) is

The number of words formed using the letters of the word MATHEMATICS in which the two M's are separated but the two I's are not together (or similar arrangement problem giving answer 84) is:

In a plane, there are 5 straight lines which pass through a given point, 6 others which all pass through another given point, and 7 others which all pass through a third given point. Supposing no three lines intersect at any point and no two are parallel, find the number of triangles formed by the intersection of the straight line.

How many 4-digit numbers between 2000 and 5000 are multiples of 3, formed using the digits 0, 1, 2, 3, 4 (repetition not allowed)?

Find the number of ways in which India can win the series of 11 matches (If no match is drawn and all matches are played).

Number of ways in which 25 identical things be distributed among five persons if each gets odd number of things is

Man members of the 1st team can be selected in 15 ways and woman members of the 1st team can be selected in 15 ways. Teams are to be formed such that the number of ways decreases by 1 for each subsequent team. Find the total number of ways of selecting 15 teams.

The total number of ways in which three distinct numbers in A.P. can be selected from the set \(\{1, 2, 3, \ldots, 24\}\) is equal to

How many 10-digit numbers can be made with odd digits so that no two consecutive digits are the same?

Let \(A\) and \(B\) be two sets containing 2 elements and 4 elements, respectively. The number of subsets of \(A \times B\) having 3 or more elements is

A person tries to form as many different parties as he can out of his 20 friends. Each party should consist of the same number. How many friends should be invited at a time? In how many of these parties would the same friends be found?

A seven-digit number without repetition and divisible by 9 is to be formed by using seven digits out of 1, 2, 3, 4, 5, 6, 7, 8, 9. The number of ways in which this can be done is

A variable name in certain computer language must be either a alphabet or a alphabet followed by a decimal digit. Find the total number of different variable names that can exist in that language.

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

Number of ways equals number of solutions of \(x_1 + x_2 + x_3 = 8\); \(x_i \geq 1\) or \((x_1 - 1) + (x_2 - 1) + (x_3 - 1) = 5\); \(x_i \geq 1\) or \(y_1 + y_2 + y_3 = 5\); \(y_i \geq 0\), which is \({}^{5+2}C_2 = \dfrac{7 \times 6}{2} = 21\). The number of ways is:

Among the \(8!\) permutations of the digits 1, 2, 3, ..., 8, consider those arrangements which have the following property. If we take any five consecutive positions, the product of the digits in these positions is divisible by 5. The number of such arrangements is equal to

The total number of divisors of 480 that are of the form \(4n + 2\), \(n \geq 0\), is equal to

A double-decker bus carry \((u + \ell)\) passengers, \(u\) in the upper deck and \(\ell\) in the lower deck. Find the number of ways in which the \((u + \ell)\) passengers can be distributed in the two decks, if \(r\ (\leq \ell)\) particular passengers refuse to go in the upper deck and \(s\ (\leq u)\) refuse to sit in the lower deck.

If \(\alpha = {}^mC_2\), then \({}^\alpha C_2\) is equal to

Here, \(a_r = {}^{2n}C_r\). Find the value of \(\displaystyle\prod_{r=1}^{2n}\left(1+\dfrac{a_r}{a_{r-1}}\right)\).

Let \(A = \{x_1, x_2, \ldots, x_7\}\) and \(B = \{y_1, y_2, y_3\}\) be two sets containing seven and three distinct elements respectively. Then the total number of functions \(f: A \to B\) that are onto, if there exists exactly three elements \(x\) in \(A\) such that \(f(x) = y_2\), is equal to

(a) If \({}^{22}P_{r+1} : {}^{20}P_{r+2} = 11:52\), find \(r\).

Let \(A = \{1, 2, 3, 4, 5, 6, 7\}\). The number of surjective functions defined from \(A\) to \(A\) such that \(f(i) = i\) for at least four values of \(i\) from \(i = 1, 2, \ldots, 7\) is:

The number of four-digit numbers that can be made with the digits 1, 2, 3, 4, and 5 in which at least two digits are identical is