Set Solving - 2

Algebra Level pending

Let there be three subsets of the set X X containing n n elements.

What is the probability that the intersection of the 3 subsets is a singleton set (i.e. contains only 1 element)? (For n = 2018 n=2018 )

If the answer is of the form 2018 ( a b ) c d \frac {2018(a^b)}{c^d} , where a , b , c , d a,b,c,d are integers, find a + b + c + d a+b+c+d .

(Note: a , c a,c are the smallest they could be, and b , d b,d are the largest they could be)

Bonus : Can you do this for any m m number of subsets having exactly r r elements common?


The answer is 4050.

Parth Sankhe by Parth Sankhe , 27, India

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

1 solution

Parth Sankhe
Nov 17, 2018

The answer is n ⋅ 7 n − 1 8 n \frac {n\cdot 7^{n-1}}{8^n}

Thus, a=7, b=2017, c=8, d=2018

No. of ways to select that one common element = n

No. of regions where the other n-1 could go = 7

(There are 8 regions created when there are three subsets of a set; think of this as a classic 3 circled Venn diagram, and you can count the number of regions yourself → \rightarrow [just A], [just B], [just C], [just A and B], [just B and C], [just A and C], [A and B and C], and [neither of the three].)

Therefore, number of ways there could be only one common element = n 7 n − 1 n7^{n-1}

In general, the probability that the intersection of m m sets has exactly r r elements in it = ⁿ C r ( 2 m − 1 ) n − r 2 m n \frac {ⁿC_r(2^m-1)^{n-r}}{2^{mn}}

0 Helpful 0 Interesting 0 Brilliant 0 Confused

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...