Question

If ${}^9{P_5} + 5 \cdot {}^9{P_4} = {}^{10}{P_r}$. Find $r$.

Answer 1

Here we need to find the value of the given variable. Here we will first apply the formula of the permutation to simplify the equation. Then we will apply the mathematical operations that will be required to simplify the given expression further. After simplifying the terms, we will get the value of the required variable.

Formula used:
${}^n{P_r} = \dfrac{{n!}}{{\left( {n - r} \right)!}}$, $r$ is the number of objects selected and $n$ total number of elements in the set.

Complete step-by-step answer:
Here we need to find the value of the given variable.
The given expression is
${}^9{P_5} + 5 \cdot {}^9{P_4} = {}^{10}{P_r}$
Now, we will use this formula of permutation ${}^n{P_r} = \dfrac{{n!}}{{\left( {n - r} \right)!}}$ in the above equation. Therefore, we get
$\Rightarrow \dfrac{{9!}}{{\left( {9 - 5} \right)!}} + 5 \cdot \dfrac{{9!}}{{\left( {9 - 4} \right)!}} = \dfrac{{9!}}{{\left( {9 - r} \right)!}}$
Subtracting the terms in the denominator, we get
$\Rightarrow \dfrac{{9!}}{{4!}} + 5 \cdot \dfrac{{9!}}{{5!}} = \dfrac{{9!}}{{\left( {9 - r} \right)!}}$
Dividing both side by $9!$, we get
$\Rightarrow \dfrac{1}{{4!}} + 5 \cdot \dfrac{1}{{5!}} = \dfrac{1}{{\left( {9 - r} \right)!}}$
$\Rightarrow \dfrac{1}{{4!}} + 5 \times \dfrac{1}{{5 \times 4!}} = \dfrac{{10}}{{\left( {9 - r} \right)!}}$
On multiplying the terms, we get
$\Rightarrow \dfrac{1}{{4!}} + \dfrac{1}{{4!}} = \dfrac{{10}}{{\left( {9 - r} \right)!}}$
On adding the numbers, we get
$\Rightarrow \dfrac{2}{{4!}} = \dfrac{{10}}{{\left( {9 - r} \right)!}}$
Dividing both sides by 2, we get
$\Rightarrow \dfrac{1}{{4!}} = \dfrac{5}{{\left( {9 - r} \right)!}}$
On cross multiplying the terms, we get
$\begin{array}{l} \Rightarrow \left( {9 - r} \right)! = 5 \times 4!\\\ \Rightarrow \left( {9 - r} \right)! = 5!\end{array}$
On equating the two factorials, we get
$\Rightarrow 9 - r = 5$
On subtracting the number 9 form sides, we get
$\Rightarrow 9 - r - 9 = 5 - 9$
On further simplification, we get
$\Rightarrow - r = - 4$
On multiplying the number $- 1$ to both sides, we get
$\Rightarrow - 1 \times - r = - 1 \times - 4$
On further simplification, we get
$\Rightarrow r = 4$
Hence, the required value of the variable $r$ is equal to 4.

Note: Here we have obtained the value of the variable used in the expression. We have also used the formula of permutation here. Here, ${}^n{P_r}$ means the number of ways to choose $r$ different things from the set of $n$ things without replacement. Here we need to know the method of finding the factorial of any number. Factorial is defined as the multiplication of all the whole numbers from the given number down to the number 1.