Question

║⊕QUESTION⊕║
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CLASS 11
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2 finite sets have m and n elements. The total number of subsets of the first set is 48 more than the total number of subsets of the second set. Find the values of m and n.

Answer 1

➫Question :-

2 finite sets have m and n elements. The total number of subsets of the first set is 48 more than the total number of subsets of the second set. Find the values of m and n.

➫Answer :-

→ m = 6 and n = 4 .

➫To Find :-

→ Value of m and n .

➫Solution :-

Let ,

$\rm x \: and \: y \: are \: two \: sets \: having \: number \: of \: \\ \rm elements \: m \: and \: n \: respectively \: \\ \\ \because \rm there \: is \: 2 \: finite \: set \: \\ \therefore \: \\ \to \rm number \: of \: subset \: of \: x = {2}^{m} \\ \: \\ \to \rm \: number \: of \: subset \: of \: y = {2}^{n} \: \\ \\ \bf \red{ According \: to \: the \: question \: } \\ \\ \rm The \: total \: number \: of \: subsets \: of \: the \: first \: \\ \rm set \: is \: 48 \: more \: than \: the \: total \: number \: of \: \\ \rm subsets \: of \: the \: second \: set. \: \\ \\ \to \rm {2}^{m} = 48 + {2}^{n} \\ \\ \to \rm {2}^{m} - {2}^{n} = 48 \\ \\ \to \rm {2}^{n} \{ \frac{ {2}^{m} }{ {2}^{n} } - 1 \} = {2}^{4} \{ {2}^{2} - 1 \} \\ \\ \to \rm {2}^{n} \{ {2}^{m - n} - 1 \} = {2}^{4} \{ {2}^{2} - 1 \} \\ \\ \rm \green{ on \: comparing \: between} \red{ \: both \: sides} \\ \\ \to \rm n \: = 4 \: \: and \: \: m - n = 2 \\ \\ \bf so \: \\ \to \rm \: m - 4 = 2 \\ \\ \to \rm \: m = 2 + 4 \\ \\ \to \boxed{ \rm m = 6 \: \: and \: n \: = 4} \\ \\ \\ \bf hence \\ \\ \to \rm number \: of \: subset \: of \: {1}^{st} \: set \: are \\ \: \: \: \: \: \: \to \: \rm \: {2}^{m} = {2}^{6} = 64 \\ \\ \bf and \: \\ \rm number \: of \: subset \: of \: {2}^{nd} \: set \: are \: \\ \: \: \: \: \: \to \rm {2}^{n} = {2}^{4} = 16$

Hence ,

The total number of subsets of the first set is 48 more than the total number of subsets of the second set.

→ 64 = 48 + 16 .

Answer 2

Step-by-step explanation:

Two finite sets have m and n elements. If the total number of subsets of first set is 56 more than the total number of subsets of second set, then find the values of m and n.

ANSWER

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.

thank you so much for giving opportunity to answer.....

I guess it helped u