Find the number of ways in which 8 distinct toys can be distributed among 5 childrens.
[6] Permutations and Combinations
Chapter: [6] Permutations and Combinations
Concept: undefined >> undefined
Find the number of ways in which one can post 5 letters in 7 letter boxes ?
[6] Permutations and Combinations
Chapter: [6] Permutations and Combinations
Concept: undefined >> undefined
\[\frac{n^{11}}{11} + \frac{n^5}{5} + \frac{n^3}{3} + \frac{62}{165}n\] is a positive integer for all n ∈ N.
[6] Principle of Mathematical Induction
Chapter: [6] Principle of Mathematical Induction
Concept: undefined >> undefined
Three dice are rolled. Find the number of possible outcomes in which at least one die shows 5 ?
[6] Permutations and Combinations
Chapter: [6] Permutations and Combinations
Concept: undefined >> undefined
Find the total number of ways in which 20 balls can be put into 5 boxes so that first box contains just one ball ?
[6] Permutations and Combinations
Chapter: [6] Permutations and Combinations
Concept: undefined >> undefined
In how many ways can 5 different balls be distributed among three boxes?
[6] Permutations and Combinations
Chapter: [6] Permutations and Combinations
Concept: undefined >> undefined
In how many ways can 7 letters be posted in 4 letter boxes?
[6] Permutations and Combinations
Chapter: [6] Permutations and Combinations
Concept: undefined >> undefined
\[\frac{1}{2}\tan\left( \frac{x}{2} \right) + \frac{1}{4}\tan\left( \frac{x}{4} \right) + . . . + \frac{1}{2^n}\tan\left( \frac{x}{2^n} \right) = \frac{1}{2^n}\cot\left( \frac{x}{2^n} \right) - \cot x\] for all n ∈ N and \[0 < x < \frac{\pi}{2}\]
[6] Principle of Mathematical Induction
Chapter: [6] Principle of Mathematical Induction
Concept: undefined >> undefined
In how many ways can 4 prizes be distributed among 5 students, when
(i) no student gets more than one prize?
(ii) a student may get any number of prizes?
(iii) no student gets all the prizes?
[6] Permutations and Combinations
Chapter: [6] Permutations and Combinations
Concept: undefined >> undefined
There are 10 lamps in a hall. Each one of them can be switched on independently. Find the number of ways in which the hall can be illuminated ?
[6] Permutations and Combinations
Chapter: [6] Permutations and Combinations
Concept: undefined >> undefined
Let P(n) be the statement : 2n ≥ 3n. If P(r) is true, show that P(r + 1) is true. Do you conclude that P(n) is true for all n ∈ N?
[6] Principle of Mathematical Induction
Chapter: [6] Principle of Mathematical Induction
Concept: undefined >> undefined
\[\frac{(2n)!}{2^{2n} (n! )^2} \leq \frac{1}{\sqrt{3n + 1}}\] for all n ∈ N .
[6] Principle of Mathematical Induction
Chapter: [6] Principle of Mathematical Induction
Concept: undefined >> undefined
\[1 + \frac{1}{4} + \frac{1}{9} + \frac{1}{16} + . . . + \frac{1}{n^2} < 2 - \frac{1}{n}\] for all n ≥ 2, n ∈ N
[6] Principle of Mathematical Induction
Chapter: [6] Principle of Mathematical Induction
Concept: undefined >> undefined
Evaluate each of the following:
8P3
[6] Permutations and Combinations
Chapter: [6] Permutations and Combinations
Concept: undefined >> undefined
Evaluate each of the following:
10P4
[6] Permutations and Combinations
Chapter: [6] Permutations and Combinations
Concept: undefined >> undefined
x2n−1 + y2n−1 is divisible by x + y for all n ∈ N.
[6] Principle of Mathematical Induction
Chapter: [6] Principle of Mathematical Induction
Concept: undefined >> undefined
Evaluate each of the following:
6P6
[6] Permutations and Combinations
Chapter: [6] Permutations and Combinations
Concept: undefined >> undefined
Evaluate each of the following:
P(6, 4)
[6] Permutations and Combinations
Chapter: [6] Permutations and Combinations
Concept: undefined >> undefined
\[\sin x + \sin 3x + . . . + \sin (2n - 1)x = \frac{\sin^2 nx}{\sin x}\]
[6] Principle of Mathematical Induction
Chapter: [6] Principle of Mathematical Induction
Concept: undefined >> undefined
\[\text{ Prove that } \cos\alpha + \cos\left( \alpha + \beta \right) + \cos\left( \alpha + 2\beta \right) + . . . + \cos\left[ \alpha + \left( n - 1 \right)\beta \right] = \frac{\cos\left\{ \alpha + \left( \frac{n - 1}{2} \right)\beta \right\}\sin\left( \frac{n\beta}{2} \right)}{\sin\left( \frac{\beta}{2} \right)} \text{ for all n } \in N .\]
[6] Principle of Mathematical Induction
Chapter: [6] Principle of Mathematical Induction
Concept: undefined >> undefined
