Question

Two finite sets have p and q elements respectivelt. The total number of subsets of first set id 224 more than the total number of subsets of the second set. Find the value of p and q.

Answer 1

Given :  Two finite sets have p and q elements respectively. The total number of subsets of first set is 224 more than the total number of subsets of the second set

To find :  value of p and q

Solution:

total number of subsets of first set is 224 more than the total number of subsets of the second set

=>  $2^p - 2^q = 224$  

=> let say p = q + x

=> $2^{(q+x)} - 2^q = 224$

=> $2^q(2^x - 1) = 224$

2ˣ - 1  Can not have 2 as factor  

224 = 2 * 2 * 2 * 2* 2 * 7

224  = 2⁵ * 7    

=> 2ˣ - 1  = 7   => 2ˣ =  8  

=> 2ˣ =  2³  

& q = 5

p = q + x  = 5 + 3  =  8

p = 8

& q = 5

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