Permutations and Combinations

The number of ways to distribute 10 identical red pens and 14 identical blue pens among four persons such that each person gets 6 pens, is ______________.

Let $S = \{1, 2, 3, \ldots, 10\}$. Consider the set $X = \{R : R \text{ is an equivalence relation on the set } S \text{ such that } R \text{ has exactly 42 elements}\}$. Then the number of elements in $X$ is ____________.

Let $S$ be the set of all seven-digit numbers that can be formed using the digits $0, 1$ and $2$. For example, $2210222$ is in $S$, but $0210222$ is NOT in $S$.Then the number of elements $x$ in $S$ such that at least one of the digits $0$ ...

Let the set of all relations $R$ on the set $\{a, b, c, d, e, f\}$, such that $R$ is reflexive and symmetric, and $R$ contains exactly $10$ elements, be denoted by $\mathcal{S}$.Then the number of elements in $\mathcal{S}$ is ______________...

If the value of $n(Y)+n(Z)$ is $k^2$, then $|k|$ is _________.

If $n(X)={ }^m C_6$, then the value of $m$ is _____

A group of 9 students, $s_1, s_2, \ldots, s_9$, is to be divided to form three teams $X, Y$, and $Z$ of sizes 2,3 , and 4 , respectively. Suppose that $s_1$ cannot be selected for the team $X$, and $s_2$ cannot be selected for the team $Y$....

Consider 4 boxes, where each box contains 3 red balls and 2 blue balls. Assume that all 20 balls are distinct. In how many different ways can 10 balls be chosen from these 4 boxes so that from each box at least one red ball and one blue bal...

The number of 4-digit integers in the closed interval [2022, 4482] formed by using the digits \(0,2,3,4,6,7\) is _________.

Let \({S_1} = \left\{ {(i,j,k):i,j,k \in \{ 1,2,....,10\} } \right\}\),\({S_2} = \left\{ {(i,j):1 \le i < j + 2 \le 10,i,j \in \{ 1,2,...,10\} } \right\}\),$${S_3} = \left\{ {(i,j,k,l):1 \le i < j < k < l,i,j,k,l \in \{ 1,2,...,10\} } \righ...

In a hotel, four rooms are available. Six persons are to be accommodated in these four rooms in such a way that each of these rooms contains at least one person and at most two persons. Then the number of all possible ways in which this can...

An engineer is required to visit a factory for exactly four days during the first 15 days of every month and it is mandatory that no two visits take place on consecutive days. Then the number of all possible ways in which such visits to the...

Let |X| denote the number of elements in a set X. Let S = {1, 2, 3, 4, 5, 6} be a sample space, where each element is equally likely to occur. If A and B are independent events associated with S, then the number of ordered pairs (A, B) such...

Five persons A, B, C, D and E are seated in a circular arrangement. If each of them is given a hat of one of the three colours red, blue and green, then the number of ways of distributing the hats such that the persons seated in adjacent se...

In a high school, a committee has to be formed from a group of 6 boys M1, M2, M3, M4, M5, M6 and 5 girls G1, G2, G3, G4, G5.(i) Let \(\alpha\)1 be the total number of ways in which the committee can be formed such that the committee has 5 ...

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 .................

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 lette...

A debate club consists of 6 girls and 4 boys. A team of 4 members is to be select from this club including the selection of a captain (from among these 4 members ) for the team. If the team has to include at most one boy, then the number of...

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 ...

Six cards and six envelopes are numbered 1, 2, 3, 4, 5, 6 and cards are to be placed in envelopes so that each envelope contains exactly one card and no card is placed in the envelope bearing the same number and moreover the card numbered 1...

Let \({n_1}\, < {n_2}\, < \,{n_3}\, < \,{n_4}\, < {n_5}\) be positive integers such that \({n_1}\, + {n_2}\, + \,{n_3}\, + \,{n_4}\, + {n_5}\) = 20. Then the number of such destinct arrangements $$\,({n_1}\,,\,{n_2},\,\,{n_3},\,\,{n_4}\,,{n...

Let \({n \ge 2}\) be an integer. Take n distinct points on a circle and join each pair of points by a line segment. Colour the line segment joining every pair of adjacent points by blue and the rest by red. If the number of red and blue lin...

Consider the set of eight vectors \(V = \left\{ {a\,\hat i + b\,\hat j + c\hat k:a,\,b,\,c\, \in \left\{ { - 1,\,1} \right\}} \right\}\). Three non-coplanar vectors can be chosen from v in \({2^p}\) ways. Then p is

Let \({{a_n}}\) denote the number of all n-digit positive integers formed by the digits 0, 1 or both such that no consecutive digits in them are 0.Let \({{b_n}}\) = the number of such n-digit integers ending with digit 1 and \({{c_n}}\) =t...

Let \({{a_n}}\) denote the number of all n-digit positive integers formed by the digits 0, 1 or both such that no consecutive digits in them are 0.Let \({{b_n}}\) = the number of such n-digit integers ending with digit 1 and \({{c_n}}\) =t...

The total number of ways in which 5 balls of different colours can be distributed among 3 persons so that each person gets at least one ball is

Let \(\left( {x,\,y,\,z} \right)\) be points with integer coordinates satisfying the system of homogeneous equation:
\($\matrix{ {3x - y - z = 0} \cr { - 3x + z = 0} \cr { - 3x + 2y + z = 0} \cr }\)$
Then the number of su...

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

Consider all possible permutations of the letters of the word ENDEANOEL. Match the Statements/Expressions in Column I with the Statements/Expressions in Column II.

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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 :

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

If the LCM of p, q is \({r^2}\,{r^4}\,{s^2}\), where r, s, t are prime numbers and p, q are the positive integers then number of ordered pair (p, q) is

A rectangle with sides of lenght (2m - 1) and (2n - 1) units is divided into squares of unit lenght by drawing parallel lines as shown in the diagram, then the number of rectangles possible with odd side lengths is

If total number of runs scored in n matches is \(\left( {{{n + 1} \over 4}} \right)\,\,({2^{n + 1}} - n - 2)\,\) where \(n > 1\), and the runs scored in the \({k^{th}}\) match are given by k. \(\,{2^{n + 1 - k}}\), where \(1 \le k \le n\). ...

Prove by permulation or otherwise \({{({n^2})!} \over {{{(n!)}^n}}}\) is an integer \((n \in {1^ + })\).

The number of arrangements of the letters of the word BANANA in which the two N's do not appear adjacently is

Let \({T_n}\) denote the number of triangles which can be formed using the vertices of a regular polygon of n sides. If \({T_{n + 1}} - {T_n} = 21\), then n equals

How many different nine digit numbers can be formed from the number 223355888 by rearranging its digits so that the odd digits occupy even positions?

An n-digit number is a positive number with exactly digits. Nine hundred distinct n-digit numbers are to be formed using only the three digits 2, 5 and 7. The smallest value of n for which this is possible is

A committee of 12 is to be formed from 9 women and 8 men. In how many ways this can be done if at least five women have to included in a committee? In how many of these committees? In how may of these committees
(a) The women are in majo...

Eighteen guests have to be seated, half on each side of a long table. Four particular guests desire to sit on one particular side and three others on the other side. Determine the number of ways in which the sitting arrangements can be made...

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

There are four balls of different colours and four boxes of colours, same as those of the balls. The number of ways in which the balls, one each in a box, could be placed such that a ball does not go to a box of its own colour is..............

Total number of ways in which six ' + ' and four ' - ' signs can be arranged in a line such that no two ' - ' signs occur together is.....................................

A box contains two white balls, three black balls and four red balls. In how many ways can three balls be drawn from the box if at least one black ball is to be included in the draw?

The product of any r consecutive natural numbers is always divisible by r!

7 relatives of a man comprises 4 ladies and 3 gentlemen ; his wife has also 7 relatives ; 3 of them are ladies and 4 gentlemen. In how many ways can they invite a dinner party of 3 ladies and 3 gentlemen so that there are 3 of man's relativ...

The side AB, BC and CA of a triangle ABC have 3, 4 and 5 interior points respectively on them. The number of triangles that can be constructed using these interior points as vertices is .......................

m men and n women are to be seated in a row so that no two women sit together. If \(m > n\), then show that the number of ways in which they can be seated is \(\,{{m!(m + 1)!} \over {(m - n + 1)!}}\)

Eight chairs are numbered 1 to 8. Two women and three men wish to occupy one chair each. First the women choose the chairs from amongst the chairs marked 1 to 4; and then the men select the chairs from amongst the remaining. The number of p...