Question

10pr is equal to 7pr+2 find r ​

Answer 1

The value of r = 4

Correct question :

$\displaystyle \sf{ {}^{10} P_r = {}^{7}P_{r + 2} } \: \: \: find \: \: r$

Given :

$\displaystyle \sf{ {}^{10} P_r = {}^{7}P_{r + 2} }$

To find :

The value of r

Solution :

Step 1 of 2 :

Write down the given equation

Here the given equation is

$\displaystyle \sf{ {}^{10} P_r = {}^{7}P_{r + 2} }$

Step 2 of 2 :

Find the value of r

$\displaystyle \sf{ {}^{10} P_r = {}^{7}P_{r + 2} }$

$\displaystyle \sf{ \implies \frac{10!}{(10 - r)!} = \frac{7!}{(7- r - 2)!}}$

$\displaystyle \sf{ \implies \frac{10!}{(10 - r)!} = \frac{7!}{(5- r )!}}$

$\displaystyle \sf \implies \frac{10 \times 9 \times 8 \times 7!}{(10 - r)(9- r )(8- r )(7- r )(6- r ) \times (5- r )!} = \frac{7!}{(5- r )!}$

$\displaystyle \sf \implies \frac{10 \times 9 \times 8 }{(10 - r)(9- r )(8- r )(7- r )(6- r ) } = 1$

$\displaystyle \sf \implies (10 - r)(9- r )(8- r )(7- r )(6- r ) = 10 \times 9 \times 8$

$\displaystyle \sf \implies (10 - r)(9- r )(8- r )(7- r )(6- r ) = 6 \times 5 \times 4 \times 3 \times 2$

Comparing both sides we get

10 - r = 6

⇒ r = 10 - 6

⇒ r = 4

Hence the required value of r = 4

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