Question

Two finite sets have 'm' and 'n' elements respectively. The total number of subsets of the first set is 192 more than the total number of subsets of the second set. The values of 'm' and 'n' respectively are
​

Answer 1

Step-by-step explanation:

Let A has m elements

Let B has n elements

Total number of students of A=2

m

Total number of students of B=2

n

It is given ⇒2

m

−2

n

=56

2

n

(2

m−n

−1)=56

⇒2

n

=even and 2

m−n

−1=0 odd

Now,

56=8×7=2

3

×2

7

⇒2

n

(2

m−n

−1)=2

3

×7

⇒n=3

Now, 8(2

m−3

−1)=8×7

⇒2

m−3

−1=7

⇒2

m−3

=8=2

3

⇒m−3=3

⇒m=6.

Answer By gentryamansharma

Answer 2

Given : Two finite sets have 'm' and 'n' elements respectively.

The total number of subsets of the first set is 192 more than the total number of subsets of the second set.  

To Find : The values of 'm' and 'n' respectively  

​

Solution:

Two finite sets have 'm' and 'n' elements respectively.

Number of subsets  $2^m$ and $2^n$  respectively

$2^m=2^n+192$

$2^m-2^n=192$

$2^n(2^{m-n}-1)=192$

192 = 2 * 2 * 2 * 2 * 2 * 2 * 3

= 2⁶ (3)

=  2⁶ (4-1)

= 2⁶(2² - 1)

comparing  with

$2^n(2^{m-n}-1)$

n = 6

m - n = 2

=> m = 8

2⁸  -2⁶ = 256 - 64  = 192

The values of 'm' and 'n' respectively are  8 and 6

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