   Question Answers

# The total number of permutation of n different things taken not more than r at a time, when each thing may be repeated any number of times is$\left( \text{a} \right)\text{ }{{n}^{r}}-1$$\left( \text{b} \right)\text{ }\dfrac{n\left( {{n}^{r}}-1 \right)}{n-1}$$\left( \text{c} \right)\text{ }\dfrac{{{n}^{r}}-1}{n-1}$$\left( \text{d} \right)\text{ }\dfrac{n\left( {{n}^{r}}-1 \right)}{n+1}$  Hint: To solve the given question, we will make use of the fact that the minimum number of things which are taken out is 1 and the maximum number of things taken at a time is r. Here, we will assume that x things are taken out from n things such that $1\le x\le r.$ Then, we will make use of the fact that if we have to choose x things from n things and there are as many repetitions of these things as we want, then the number of ways of doing this will be ${{n}^{x}}.$ Then, we will put different values of x from 1 to r and then we will add all these. On adding, we will get a series whose sum will be found out by using the appropriate formula.

Complete step by step solution:
To start with, we are given that we cannot take more than r things out of n at a time i.e. the maximum number of things which can be taken out at a time is r. Let us assume that we have taken out x things out of n things such that $1\le x\le r.$ Now, we are given that when we take one thing out, the number of distinct things remains the same, i.e. n. This means that things have a sufficient number of repetitions. Thus, when we will take one thing, next time, we will still have n distinct things. Now, we know that x varies from 1 to r. Let x = 1, i.e. we take out one thing at a time. The number of ways of doing this is n.
Now, we assume that x = 2. This means that we have to select two things at a time. The number of ways of selecting the first thing is n. Now, we know that there are enough repetitions of things such that even after taking one thing, the number of distinct things remains n. Now, the number of ways of selecting the second thing is n. Thus, the total number of permutations become $n\times n={{n}^{2}}.$
Similarly, when n = 3, the total number of ways of selecting the first, second, and third things are n, n, and n respectively. Thus, the total number of permutations become $n\times n\times n={{n}^{3}}.$ Thus, when x = r, the total number of permutations will be ${{n}^{r}}.$
Thus, the overall number of permutations will be the sum of all these cases. Thus,
$\text{Total permutations}=n+{{n}^{2}}+{{n}^{3}}+......+{{n}^{r}}$
We can see that the above series is a geometric progression with common ratio n. We know that the sum of the terms of a GP is equal to $a\left( \dfrac{1-{{p}^{q}}}{1-p} \right)$ where a is the first term, p is the common ratio and q are the total number of terms. Thus, we will get,
$\Rightarrow \text{Total permutation}=\dfrac{n\left( 1-{{n}^{r}} \right)}{1-n}$
$\Rightarrow \text{Total permutation}=\dfrac{n\left( {{n}^{r}}-1 \right)}{n-1}$
Hence, option (b) is the right answer.

Note: In the question, it is given that each thing may be repeated at any number of times. In our case, each thing should be repeated at least r times because when we have selected r things at a time, then it is possible that we have selected each thing of the same kind.

View Notes
Different Formats Of Writing A Resume  To Determine the Mass of Two Different Objects Using a Beam Balance  Derivation of Reynolds Number  Number of Moles Formula  CBSE Class 11 Physics Law of Motion Formulas  CBSE Class 11 Physics Thermal Properties of Matter Formulas  Table of 11 - Multiplication Table of 11  CBSE Class 11 Physics Mechanical Properties of Solids Formulas  CBSE Class 11 Physics Systems of Particles and Rotational Motion Formulas  CBSE Class 11 Physics Mechanical Properties of Fluids Formulas  Important Questions for CBSE Class 11 Maths Chapter 7 - Permutations and Combinations  Important Questions for CBSE Class 6 English Honeysuckle Chapter 5 - A Different Kind of School  Important Questions for CBSE Class 11 Biology Chapter 8 - Cell The Unit of Life  Important Questions for CBSE Class 11 English Snapshots Chapter 1 - The Summer of the Beautiful White Horse  Important Questions for CBSE Class 11 Maths Chapter 4 - Principle of Mathematical Induction  CBSE Class 8 Science Reaching The Age of Adolescence Worksheets  Important Questions for CBSE Class 11 Physics Chapter 11 - Thermal Properties of Matter  NCERT Books Free Download for Class 11 Maths Chapter 7 - Permutations and Combinations  Important Questions for CBSE Class 11 Indian Economic Development Chapter 1 - Indian Economy on the Eve of Independence  Important Questions for CBSE Class 11 Accountancy Chapter 8 - Bill Of Exchange  Maths Question Paper for CBSE Class 12 - 2016 Set 1 N  Chemistry Question Paper for CBSE Class 12 - 2016 Set 1 N  Previous Year Question Paper for CBSE Class 12 Physics - 2016 Set 1 N  Previous Year Question Paper of CBSE Class 10 English  CBSE Class 12 Maths Question Paper 2020  CBSE Class 10 Maths Question Paper 2020  Maths Question Paper for CBSE Class 10 - 2011  Maths Question Paper for CBSE Class 10 - 2008  CBSE Class 10 Maths Question Paper 2017  Maths Question Paper for CBSE Class 10 - 2012  NCERT Solutions for Class 11 Maths Chapter 7  NCERT Exemplar for Class 11 Maths Chapter 7 - Permutations and Combinations (Book Solutions)  NCERT Solutions for Class 11 Maths Chapter 7 Permutations and Combinations in Hindi  NCERT Solutions Class 11 English Woven Words Prose Chapter 4 The Adventure of the Three Garridebs  NCERT Solutions for Class 11 Maths Chapter 7 Permutations and Combinations (Ex 7.4) Exercise 7.4  NCERT Solutions for Class 11 Maths Chapter 7 Permutations and Combinations (Ex 7.2) Exercise 7.2  NCERT Solutions for Class 11 Maths Chapter 7 Permutations and Combinations (Ex 7.1) Exercise 7.1  NCERT Solutions for Class 11 English Snapshots Chapter 6 - The Ghat of the Only World  NCERT Solutions for Class 11 English Hornbill Chapter 4 - Landscape of the Soul  NCERT Solutions for Class 11 Maths Chapter 7 Permutations and Combinations (Ex 7.3) Exercise 7.3  