Question

If the numbers of subsets of {1, 2, 3, 4, 5, 6, 7, 8, 9} that are subsets of neither {1, 2, 3, 4, 5} nor {4, 5, 6, 7, 8, 9} is K, then find the sum of digits of K.

Answer 1

Answer:

06

Step-by-step explanation:

Number of subsets of {1, 2, 3, 4, 5, 6, 7, 8, 9} = 29

Number of subsets of {1, 2, 3, 4,5} = 25

Number of subsets of {4, 5, 6, 7, 8, 9} = 26

Number of subsets of {4, 5} = 22

∴ Total number of subsets  

   = 512 – 92

   = 420

∴ Sum of digits = 4 + 2 + 0 = 06

Answer 2

If the numbers of subsets of {1, 2, 3, 4, 5, 6, 7, 8, 9} that are subsets of neither {1, 2, 3, 4, 5} nor {4, 5, 6, 7, 8, 9} is K, then the sum of digits of K is 9

Step-by-step explanation:

Let the sets be

A =  {1, 2, 3, 4, 5, 6, 7, 8, 9}

B = {1, 2, 3, 4, 5}

C = {4, 5, 6, 7, 8, 9}

Then

B ∩ C = {4, 5}

Subsets of B ∩ C = {4}, {5}, {4,5}, {}

If we exclude the null set then the subsets remaining will be {4}, {5}, {4,5}

No. of elements in A = 9

No. of subsets of A = $2^9$

No. of elements in B = 5

No. of subsets of B = $2^5$

No. of elements in C = 6

No. of subsets of C = $2^6$

If we subtract the no. of subsets of B and C from no. of subsets of A we get no. of subsets of A that are not the subsets of B or C

But the null set will be subtracted twice

Also as the subsets of B ∩ C excluding null set will be subtracted twice

Therefore, total number of subsets of A not the subsets of B or C

$K=2^9-2^5-2^6+1-3$

$\implies K=512-32-64-2$

$\implies K=414$

The sum of the digits of K

$=4+1+4=9$

Hope this answer is helpful.

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