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Maths-

General

Easy

Question

If ^(m+n)P₂ = 56 and ^m–nP₂ = 12 then (m, n) equals-

(5, 1)
(6, 2)
(7, 3)
(9, 6)

hint Hint:

$U sin g space f o r m u l a comma space e x p a n d space t h e space a b o v e space g i v e n space e q u a t i o n s P presuperscript n subscript r space equals space space fraction numerator n factorial over denominator left parenthesis n minus r right parenthesis factorial end fraction a n d space s i m p l i f y$

The correct answer is: (6, 2)

$G i v e n space colon space P presuperscript left parenthesis m plus n right parenthesis end presuperscript subscript 2 equals space 56 space a n d space space P presuperscript m minus n end presuperscript subscript 2 space equals space 12 U sin g space f o r m u l a comma space e x p a n d space t h e space a b o v e space g i v e n space e q u a t i o n s P presuperscript n subscript r space equals space space fraction numerator n factorial over denominator left parenthesis n minus r right parenthesis factorial end fraction P presuperscript left parenthesis m plus n right parenthesis end presuperscript subscript 2 equals space 56 rightwards double arrow fraction numerator left parenthesis m plus n right parenthesis factorial over denominator left parenthesis m plus n minus 2 right parenthesis factorial end fraction equals space 56 rightwards double arrow fraction numerator left parenthesis m plus n right parenthesis space left parenthesis m plus n minus 1 right parenthesis space left parenthesis m plus n minus 2 right parenthesis factorial over denominator left parenthesis m plus n minus 2 right parenthesis factorial end fraction equals space 56 rightwards double arrow left parenthesis m plus n right parenthesis space left parenthesis m plus n minus 1 right parenthesis space equals space 56 space L e t space m space plus space n space equals space t rightwards double arrow left parenthesis t right parenthesis space left parenthesis t minus 1 right parenthesis space equals space 56 space rightwards double arrow t squared space minus space t space minus 56 space equals space 0 O n space f a c t o r i z i n g space b y space s p l i t t i n g space o f space m i d d l e space t e r m s rightwards double arrow left parenthesis t minus 8 right parenthesis left parenthesis t space plus space 7 right parenthesis space equals space 0 rightwards double arrow space t space equals space 8 comma space minus 7 space left parenthesis n e g a t i v e space v a l u e space c a n n o t space b e space t a k e n right parenthesis rightwards double arrow space t space equals space 8 rightwards double arrow space m space plus space n space equals space 8 space........ space left parenthesis 1 right parenthesis$
Similarly
$P presuperscript left parenthesis m minus n right parenthesis end presuperscript subscript 2 equals space 12 rightwards double arrow fraction numerator left parenthesis m minus n right parenthesis factorial over denominator left parenthesis m minus n minus 2 right parenthesis factorial end fraction equals space 56 rightwards double arrow fraction numerator left parenthesis m minus n right parenthesis space left parenthesis m minus n minus 1 right parenthesis space left parenthesis m minus n minus 2 right parenthesis factorial over denominator left parenthesis m minus n minus 2 right parenthesis factorial end fraction equals space 12 rightwards double arrow left parenthesis m minus n right parenthesis space left parenthesis m minus n minus 1 right parenthesis space equals space 12 L e t space m minus space n space equals space a rightwards double arrow left parenthesis a right parenthesis space left parenthesis a minus 1 right parenthesis space equals space 12 space rightwards double arrow a squared space minus space a space minus 12 space equals space 0 O n space f a c t o r i z i n g space b y space s p l i t t i n g space o f space m i d d l e space t e r m s rightwards double arrow left parenthesis a minus 4 right parenthesis left parenthesis a space plus space 3 right parenthesis space equals space 0 rightwards double arrow space a space equals space 4 comma space minus 3 space left parenthesis n e g a t i v e space v a l u e space c a n n o t space b e space t a k e n right parenthesis rightwards double arrow space a space equals space 4 rightwards double arrow space m space minus space n space equals space 4 space........ space left parenthesis 2 right parenthesis$

Adding (1) and (2), we get
$2 m space equals space 12 rightwards double arrow space m space equals space 6$

Substituting m= 6 in (2), we get
$6 minus n space equals space 4 rightwards double arrow n equals 2$
Thus, If $P presuperscript left parenthesis m plus n right parenthesis end presuperscript subscript 2 equals space 56$ and $P presuperscript m minus n end presuperscript subscript 2$ then (m, n) equals- (6,2)

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