List of top Mathematics Questions on Combinatorics

The number of non-negative integer solutions of the equation \[ a + b + 2c = 22 \] is:
All the words (with or without meaning) formed using all the five letters of the word GOING are arranged as in a dictionary. Then the word OGGIN occurs at the place which is:
A student has to answer a multiple-choice question having 5 alternatives in which two or more than two alternatives are correct. Then the number of ways in which the student can answer that question is
All possible words (with or without meaning) that contain the word 'GENTLE' are formed using all the letters of the word 'INTELLIGENCE'. Then the number of words in which the word 'GENTLE' appears among the first nine positions only is
5 boys and 5 girls have to sit around a table. The number of ways in which all of them can sit so that no two boys and no two girls are together is
\( ^{20}P_3 - ^{19}P_3 = \)
In how many ways can 8 identical pens be distributed among Aal, Bal, and Cal so that Aal gets at least 1 pen, Bal gets at least 2 pens, and Cal gets at least 3 pens?
P, Q, R and S are four towns. One can travel between P and Q along 3 direct paths, between Q and S along 4 direct paths, and between P and R along 4 direct paths. There is no direct path between P and S, while there are a few direct paths between Q and R and also between R and S. One can travel from P to S either via Q, or via R, or via Q followed by R, respectively, in exactly 62 possible ways. One can also travel from Q to R either directly, or via P, or via S, in exactly 27 possible ways. Then, the number of direct paths between Q and R is:
If all the letters of the word 'HANDLE' are permuted in all possible ways and the words (with or without meaning) thus formed are arranged in dictionary order, then the rank of the word 'HELAND' is
Number of triangles whose vertices are the points \( (x, y) \) in the XY-plane with integer coordinates satisfying \( 0 \le x \le 4 \) and \( 0 \le y \le 4 \) is
From all the English alphabets, five letters are chosen and are arranged in alphabetical order. The total number of ways, in which the middle letter is ‘M’, is :
There are 12 points in a plane, no three of which are in the same straight line, except 5 points which are collinear. Then the total number of triangles that can be formed with the vertices at any three of these 12 points is:
Let P be the set of seven-digit numbers with the sum of their digits equal to 11. If the numbers in P are formed by using the digits 1, 2, and 3 only, then the number of elements in the set P is:
From a group of 7 batsmen and 6 bowlers, 10 players are to be chosen for a team, which should include at least 4 batsmen and at least 4 bowlers. One batsman and one bowler who are captain and vice-captain respectively of the team should be included. Then the total number of ways such a selection can be made, is:
The number of ways, in which the letters A, B, C, D, E can be placed in the 8 boxes of the figure below so that no row remains empty and at most one letter can be placed in a box, is:
Let $ p $ be the number of all triangles that can be formed by joining the vertices of a regular polygon $ P $ of $ n $ sides, and $ q $ be the number of all quadrilaterals that can be formed by joining the vertices of $ P $. If $ p + q = 126 $, then the eccentricity of the ellipse $ \frac{x^2}{16} + \frac{y^2}{n} = 1 $ is:
Line L1 of slope 2 and line L2 of slope \( \frac{1}{2} \) intersect at the origin O. In the first quadrant, \( P_1, P_2, ..., P_{12} \) are 12 points on line L1 and \( Q_1, Q_2, ..., Q_9 \) are 9 points on line L2. Then the total number of triangles that can be formed having vertices at three of the 22 points O, \( P_1, P_2, ..., P_{12} \), \( Q_1, Q_2, ..., Q_9 \), is:
For $ n \geq 2 $, let $ S_n $ denote the set of all subsets of $ \{1, 2, 3, \ldots, n\} $ with no two consecutive numbers. For example, $ \{1, 3, 5\} \in S_6 $, but $ \{1, 2, 4\} \notin S_6 $. Then, find $ n(S_5) $.
Let m and n, $ m<n $ be two 2-digit numbers. Then the total number of pairs (m, n) such that $ \gcd(m, n) = 6 $, is _______

If \[ \sum_{r=1}^{30} r^2 \left( \binom{30}{r} \right)^2 = \alpha \times 2^{29}, \] then \( \alpha \) is equal to _______.

If \[ \sum_{r=1}^{30} r^2 \left( \binom{30}{r} \right)^2 = \alpha \times 2^{29}, \] then \( \alpha \) is equal to _________.
A scientific committee is to be formed from 6 Indians and 8 foreigners, which includes at least 2 Indians and double the number of foreigners as Indians. Then the number of ways is :
Let \(M\) be any \(3\times3\) matrix with entries from the set {0, 1, 2}. The maximum number of such matrices, for which the sum of diagonal elements of \(M^T M\) is seven, is __________.
The students S1, S2, ......., S10 are to be divided into 3 groups A, B and C such that each group has at least one student and the group C has at most 3 students. Then the total number of possibilities of forming such groups is _________