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Maths-

General

Easy

Question

If $blank to the power of n P subscript r equals 720 comma to the power of n C subscript r equals 120$ then (n,r) is

(7,4)
(6,2)
(8,4)
(10,3)

hint Hint:

To find the value of n and r first we will use the formula $P presuperscript n subscript r equals fraction numerator n factorial over denominator left parenthesis n minus r right parenthesis factorial end fraction$ and $C presuperscript n subscript r equals fraction numerator n factorial over denominator r factorial left parenthesis n minus r right parenthesis factorial end fraction$ we will get two equations then we will equate them and get the values.

The correct answer is: (10,3)

Given, $blank to the power of n P subscript r equals 720 comma to the power of n C subscript r equals 120$
$blank to the power of n P subscript r equals 720 rightwards double arrow fraction numerator n factorial over denominator open parentheses n minus r close parentheses factorial end fraction equals 720 rightwards double arrow n factorial equals 720 left parenthesis n minus r right parenthesis factorial... space e q u space 1$
$blank to the power of n C subscript r equals 120 rightwards double arrow fraction numerator n factorial over denominator r factorial left parenthesis n minus r right parenthesis factorial end fraction equals 120 rightwards double arrow n factorial equals 120 space r factorial space left parenthesis n minus r right parenthesis factorial.... space e q u space 2$
from equ 1 and 2
720(n-r)! =120 r! (n-r)!
$rightwards double arrow$ 6=r!
$rightwards double arrow$ r=3
putting r=3 in equ 1
$fraction numerator n factorial over denominator open parentheses n minus 3 close parentheses factorial end fraction equals 720 rightwards double arrow fraction numerator n left parenthesis n minus 1 right parenthesis left parenthesis n minus 2 right parenthesis left parenthesis n minus 3 right parenthesis factorial over denominator left parenthesis n minus 3 right parenthesis factorial end fraction equals 720 rightwards double arrow n left parenthesis n minus 1 right parenthesis left parenthesis n minus 2 right parenthesis equals 10 cross times 9 cross times 8 rightwards double arrow n equals 10$

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