Two finite sets have m and n elements. The number of subsets of the first set is 224 more than that of the second set. The values of m and n are, respectively.

Question

Two finite sets have m and n elements. The number of subsets of the first set is 224 more than that of the second set. The values of m and n are, respectively. (A) 3,5 (B) 8,5 (C) 3,8 (D) 5,8

Tardigrade Admin · Accepted Answer

2m-2n=224 2n(2m-n-1)=25 × 7 2n(2m-n-1)=25(23-1) n=5 m-n=3 ⇒ m=8

Q. Two finite sets have $m$ and $n$ elements. The number of subsets of the first set is $224$ more than that of the second set. The values of $m$ and $n$ are, respectively.

3,5

8,5

3,8

5,8

$2^{m} - 2^{n} = 224$
$2^{n} (2^{m - n} - 1) = 2^{5} \times 7$
$2^{n} (2^{m - n} - 1) = 2^{5} (2^{3} - 1)$
$n = 5 m - n = 3 \Rightarrow m = 8$

Q. Two finite sets have m and n elements. The number of subsets of the first set is 224 more than that of the second set. The values of m and n are, respectively.

3,5

8,5

3,8

5,8

2m−2n=224 2n(2m−n−1)=25×7 2n(2m−n−1)=25(23−1) n=5m−n=3⇒m=8

Q. Two finite sets have $m$ and $n$ elements. The number of subsets of the first set is $224$ more than that of the second set. The values of $m$ and $n$ are, respectively.

$2^{m} - 2^{n} = 224$
$2^{n} (2^{m - n} - 1) = 2^{5} \times 7$
$2^{n} (2^{m - n} - 1) = 2^{5} (2^{3} - 1)$
$n = 5 m - n = 3 \Rightarrow m = 8$