Permutations & Combinations Questions (855)

A committee of 5 is to be chosen from a group of 9 people. The probability that a certain married couple will either serve together or not at all is

Let \(a\) and \(b\) be a variable vector such that \(a\) and \(b\) are positive integers. If \(a + b \leq 12\), then the number of values of \((a,b)\) is

Let $A=\{1,2,3,\ldots,10\}$, $B=\{4,8,12,16,20\}$ and $C=\{D:D\subseteq A,\,D\cap B\neq\emptyset\}$. The number of elements in $C$ which have at least 3 but at most 6 cardinal number 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(b) repetition of digits is not allowed?

The letters of the word COCHIN are permuted and all the permutations are arranged in an alphabetical order as in an English dictionary. The number of words that appear before the word COCHIN is

Find the number of natural numbers which are less than \(2 \times 10^8\) and which can be written by means of the digit 1 and 2.

In how many ways the number 7056 can be resolved as a product of 2 factors?

The number of 5 digit numbers which are divisible by 4, with digits from the set \(\{1, 2, 3, 4, 5\}\) and the repetition of digits is allowed, is ________.

A committee of 10 persons is to be selected from 9 women and 8 men, consisting of at least 4 women and at least 4 men. Find the number of ways the committee can be formed if Miss X and Mr. Y insist to work together.

In how many ways the sum of upper faces of four distinct dice can be six?

How many numbers of n digits can be made with the nonzero digits in which no two consecutive digits are the same?

A large pile contains red, white, green and blue balls (all alike except for colour). Find the number of ways to select 20 balls from them such that the selection has an even number of white balls.

There are tea cups with and without handles. The number of ways of selecting 2 without handle and 3 with handle is exactly 1200. What is the maximum possible number of cups in the kitchen?

Number of points with integral co-ordinates that lie inside a triangle whose co-ordinates are \((0, 0)\), \((0, 21)\) and \((21, 0)\):

The number of ways of arranging n persons, if out of any two seats located symmetrically in the middle of the row at least one is empty is

Let \(S = \{1, 2, 3, 4\}\). The total number of unordered pairs of disjoint subsets of \(S\) is equal to

There are three coplanar parallel lines. If any \(p\) points are taken on each of the lines, the maximum number of triangles with vertices on these points is

The number of seven digit integers, with sum of the digits equal to 10 and formed by using the digits 1, 2 and 3 only, is

If r, s, t are prime numbers and p, q are the positive integers such that their LCM of p, q is \(r^2t^4s^2\), then the numbers of ordered pair of (p, q) is

Find the number of factors of the number 37800. Also find the sum of the odd proper divisors of the number.

Find the total number of ways of answering five objective type questions, each question having four choices.

A library has 5 indistinguishable physics books, 4 indistinguishable mathematics books and 3 indistinguishable chemistry books. In how many distinguishable ways can a student take home 6 books?

The total number of ways of selecting six coins out of some one-rupee coins, 10 fifty-paise coins, and 7 twenty-five paise coins is

A committee of 10 persons is to be selected from 9 women and 8 men, consisting of at least 4 women and at least 4 men. Find the number of ways the committee can be formed if Miss X refuses to work with Mr. Y.

Given word: BARRACK. The total number of four-letter words that can be formed using the letters of the word BARRACK is:

Find the number of odd numbers greater than two million that can be made with the digits of the number 1315414.

The sum of the digits in the unit's place of all the 4-digit numbers formed by using the numbers 3, 4, 5 and 6, without repetition, is

The maximum number of points of intersection of five lines and four circles is

The number of different messages by signals with three dots and two dashes is

Out of 10 white, 9 black, and 7 red balls, find the number of ways in which selection of one or more balls can be made (balls of the same color are identical).

Three boys of class X, four boys of class XI, and five boys of class XII sit in a row. The total number of ways in which these boys can sit so that all the boys of same class sit together is equal to

Let \(n\) be the number of ways in which 5 boys and 5 girls can stand in a queue in such a way that all the girls stand consecutively in the queue. Let \(m\) be the number of ways in which 5 boys and 5 girls can stand in a queue in such a way that exactly four girls stand consecutively in the queue. Then the value of \(\dfrac{m}{n}\) is ________.

The number of selections of 4 letters taken from the word COLLEGE is equal to

A five-digit numbers divisible by 3 is to be formed using the numerals 0, 1, 2, 3, 4 and 5, without repetition. Find the total number of ways in which this can be done.

Consider the letters of the word MATHEMATICS. Possible number of words in which no two vowels are together is

\(2m\) white counters and \(2n\) red counters are arranged in a straight line with \((m+n)\) counters on each side of a central mark. The number of ways of arranging the counters, so that the arrangements are symmetrical with respect to the central mark is

The number of different seven-digit numbers that can be written using only the three digits 1, 2, and 3 with the condition that the digit 2 occurs twice in each number is

A postman has to deliver five letters to five different houses. Mischievously, he posts one letter through each door without looking to see if it is the correct address. In how many different ways could he do this so that exactly two of the five houses receive the correct letters.

Let A = {1, 2, 3, 4, 5, 6, 7}. The number of bijections f : A → A such that f(i) = i for at least 4 values of i is:

There are 5 mangoes and 4 apples. In how many different ways can a selection of fruits be made if (i) fruits of the same kind are different (ii) fruits of the same kind are identical?(i) Find the number of ways if fruits of the same kind are different.

There are 10 bags \(B_1, B_2, \ldots, B_{10}\) which contain 31, 32, \ldots, 40 distinct articles respectively. The total number of ways to draw 10 articles all from a single bag is

The number of ordered triplets \((a, b, c)\) from the set \(\{1, 2, 3, 4, \ldots, 100\}\) such that \(a \leq b \leq c\) is equal to

There are n married couples at a party. Each person shakes hand with every other person other than her or his spouse. The total number of handshakes must be

Find the number of divisors of 720. How many of these are even? Also find the sum of divisors.

If \(\displaystyle\sum_{r=0}^{25} \left\{ {}^{50}C_r \cdot {}^{50-r}C_{25-r} \right\} = K \cdot {}^{50}C_{25}\), then \(K\) is equal to:

n1 and n2 are four-digit numbers. Find the total number of ways of forming n1 and n2 so that n2 can be subtracted from n1 without borrowing at any stage.

How many numbers of five digits can be made with at least one repeated digit?

There are three copies each of four different books. The number of ways in which they can be arranged on a shelf is

Find number of non-negative integral solutions of the equation \(x + y + z = 10\).

All possible numbers are formed using the digits 1, 1, 2, 2, 2, 2, 3, 4, 4, taken all at a time. The number of such numbers in which the odd digits occupy even places is __________.