Permutations and Combinations

An urn contains 5 red marbles, 4 black marbles and 3 white marbles. Then the number of ways in which 4 marbles can be drawn so that the most three of them are red is(Combinations)

There are 3 sections in a question paper and each section contains 5 questions. A candidate has to answer a total of 5 questions, choosing at least one question from each section. Then the number of ways, in which the candidate can choose the questions, is:

The number of ordered pairs $(r,k)$ for which $6\cdot {}^{35}{C}_r=\left(k^2-3\right)\cdot {}^{36}{C}_{r+1}$ , where  is an integer, is:

If ⁿPᵣ = ⁿPᵣ₊₁ and ⁿCᵣ = ⁿCᵣ₋₁, then the value of r is equal to:

A scientific committee is to be formed form 6 Indians and 8 foreigners, which includes at least 2 Indians and double the number of foreigners as Indians. Then the number of ways the committee and be formed is:

  $\underset{n\to \infty }{lim}tan\left\{{\displaystyle \sum _{r=1}^{n}ta{n}^{-1}}\left(\frac{1}{1+r+r^2}\right)\right\}$ is equal to___________.

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…………….

The total number of numbers, lying between 100 and 1000 that can be formed with the digits 1, 2, 3, 4, 5, if the repetition of digits is not allowed and numbers are divisible by either 3 or 5, is…………….

The total number of positive integral solutions (x, y, z)  such that xyz = 24 is :

The number of sequences of ten terms, whose terms are either 0 or 1 or 2 , that contain exactly five 1s and exactly three 2s, is equal to