If {}ⁿP_{r} = 990 and {}ⁿC_{r} = 165 find the values of n and r

Correct Answer is : n= 11 and r = 3

Correct Answer is : n= 11 and r = 3

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Haryana Police Constable Model Paper 2

Question : 30 of 100

Marks: +1, -0

{}^{n}P_{r}

{}^{n}C_{r}

= 165 find the values of n and r

Solution:

Given:

{}^{n}P_{r}

= 990 and

{}^{n}C_{r}

= 165

Formula used:

{}^{n}P_{r}=\frac{n!}{(n-r)!}

{}^{n}C_{r}

\frac{n!}{(n-r)! \times r!}

Calculation:

{}^{n}P_{r}=\frac{n!}{(n-r)!}

….(i)

{}^{n}C_{r}

\frac{n!}{(n-r)! \times r!}

….(ii)

Divide equation (i) and (ii)

⇒

\frac{{}^{n}P_{r}}{{}^{n}C_{r}}

\frac{\frac{n}{(n-r)!}}{\frac{n!}{\frac{n}{(n-r)! \times r!}}}

\Rightarrow \frac{{}^{n}P_{r}}{{}^{n}C_{r}}

= r!

⇒

\frac{990}{165}

= r!

⇒ r! = 6

⇒ r! = 3 × 2 × 1

⇒ r = 3

The value of r putting in

{}^{n}P_{r}

=990

⇒

{}^{n}P_{3}

= 990

⇒ n × (n – 1) × (n – 2) = 11 × 10 × 9

From this we get

n = 11

∴Values of n and r are respectively 11 and 3

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