A is a set of distinct integers, and there are between

Question

A is a set of distinct integers, and there are between 300 and 400 subsets of A that contain at least one even integer. How many 3-element subsets of A are such that the sum of the 3 integers in the subset is even?

(A) 21
(B) 30
(C) 35
(D) 42
(E) 48

Show Answer

Official Answer

D

(A) 21
(B) 30
(C) 35
(D) 42
(E) 48

Jash1666 · Answer

Can anyone please provide the solution, unable to slove  kevincan   Bunuel

kevincan · Answer

If a set consists of n integers, k of which are odd, how many subsets of the set contain at least one even integer?

There are 2^n subsets in total, 2^k of which do not contain any even numbers.

kevincan · Answer

If a set consists of n integers, k of which are odd, how many subsets of the set contain at least one even integer?

There are  2^n  subsets in total. Of these,  2^k  contain no even integers, since such a subset can contain only the  k  odd integers.

Therefore, the number of subsets containing at least one even integer is

2^n-2^k .

In the problem, there are between 300 and 400 such subsets, so

300<2^n-2^k<400 .

Since  2^n>300 ,  n\ge9 . If  n=9 , then

2^9-2^k=512-2^k .

For this to be between 300 and 400,  2^k  must be 128, giving  k=7 .

Thus, the set contains 7 odd integers and  9-7=2  even integers.

For a 3-element subset to have an even sum, it must contain either:

*  \binom{2}{3}=0  (3 even integers)
*  \binom{2}{1}\binom{7}{2}=2(21)=42  (1 even and 2 odd integers)

Therefore,

42

Jash1666 · Answer

For a subset having 3 elements with an even sum, it should contain 2 odd and 1 even combination only right? Where am I going wrong

for eg : sums of 3 odds 1+3+5 is odd and 2 even 1 odd 2+4+7 is odd

kevincan · Answer

You are right ! I used AI to write out the solution and it reversed it .

HarshavardhanR · Answer

Let's first visualize the set A (set of distinct integers).

n -> number of elements (integers) in the set A, overall
k -> number of odd integers in the set A
n - k -> number of even integers in the set A

A -> { (.....k odd integers.....) (.....n-k even integers.....) }

Number of subsets of the set = \(2^n\) (this includes the null set i.e., nC0).

Number of subsets of A that contain no even integers = \(2^k\) (we have to ignore the even numbers - from the odd numbers, how many sets can be created is what will yield this).

Hence,

The number of subsets of A that contain at least one even integer = \(2^n\) - \(2^k\)

Given:

300 < \(2^n\) - \(2^k\) < 400

Analyzing this further to see what values n and k can take:

\(2^n\) > 300 + \(2^k\)

=> n >=9

Can n be 10?

\(2^n\) < 400 + \(2^k\)

1024 < 400 + \(2^k\)

\(2^k\) > 624

which means k>=10.

But this is a problem. If n = 10, k has to be lesser than 10. k>=10 is impossible.

In other words, we can see that n cannot be 10.

We can even check whether n = 11 works. Then again, we observe that k has to be >=11. Which is impossible.

Key Insight:
We already established that n>=9. But we can also see that n cannot be >9. Hence, n = 9.

So, given that n = 9,

512 > 300 + \(2^k\)

=> k <=7

512 < 400 + 2^k

=> k >=7

=> k = 7

Thus, in set A, we have 9 integers in total, out of which we have 7 odd numbers and 2 even numbers.

How many 3-element subsets of A are such that the sum of the 3 integers in the subset is even?

The only way to get an even sum is if 2 odd numbers and 1 even number are picked.

How many such subsets can be created?

7C2 x 2C1 = 42. Choice D.

---
Harsha

A is a set of distinct integers, and there are between

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

A is a set of distinct integers, and there are between

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards