Two finite sets have m and n elements. The total number of subsets of the first set is 56 more than the total number of subsets of the second set. The values of m and n are

Question

Two finite sets have m and n elements. The total number of subsets of the first set is 56 more than the total number of subsets of the second set. The values of m and n are (A) 7,6 (B) 6,3 (C) 5,1 (D) 8,7

Tardigrade Admin · Accepted Answer

No. of subsets if a set contain r elements =2r ∴ 2m-2n=56 ⇒ 2n(2m-n-1)=8 × 7=23 × 7 ∴ n=3 text and 2m-n=8=23 ⇒ m-n=3 ⇒ m-3=3 ⇒ m=6 ∴ m=6, n=3

Q. Two finite sets have $m$ and $n$ elements. The total number of subsets of the first set is $56$ more than the total number of subsets of the second set. The values of $m$ and $n$ are

$7, 6$

$6, 3$

$5, 1$

$8, 7$

No. of subsets if a set contain $r$ elements $= 2^{r}$
$∴ 2^{m} - 2^{n} = 56$
$\Rightarrow 2^{n} (2^{m - n} - 1) = 8 \times 7 = 2^{3} \times 7$
$∴ n = 3 and 2^{m - n} = 8 = 2^{3}$
$\Rightarrow m - n = 3$
$\Rightarrow m - 3 = 3$
$\Rightarrow m = 6$
$∴ m = 6, n = 3$

Q. Two finite sets have m and n elements. The total number of subsets of the first set is 56 more than the total number of subsets of the second set. The values of m and n are

7,6

6,3

5,1

8,7