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Two finite sets have m and n elements. The total number of subsets of the first set is 112 more than the total number of subsets of the second set. Find the valuesof m and n.

Given: There are two finite sets having m and n elements. 
Therefore, total possible selections from first finite set = 2m

and total possible selections from second finite set = 2n 
As, it is given that, the total no. of subsets of the first set is 112 more than that of the total no. of subsets of the second set.

It means

2m = 112 +2n 

⇒2m  - 2n =112 

 

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Number of subsets of the two sets = 2m and 2n, respectively (Formula)

According to the question,

2m = 2n – 112

ð   2m - 2n = 112

ð  2n (2m / 2n - 1) = 24 (7)

ð  2n ( 2m - n - 1) = 24 ( 7 )

2n cannot be equal to 7 (since 2n = 2 x 2 x 2 x 2 x .........)

Therefore, 2n = 24

Equating the powers of 2, we get:

n = 4.......... (1)

Also, 2m - n -1 = 7

ð  2m - n = 7+1 = 8

ð  2m - n = 23 (since 8 = 23)

Equating the powers of 2, we get:

m - n= 3

ð   m = n+3 = 4+3     ( From (1) )

ð   m = 7

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