A math teacher assigns a distinct prime number, starting with the lowe

Question

A math teacher assigns a distinct prime number, starting with the lowest, for each student in her class. If she chooses two students at random, the probability the sum of their numbers is not even is less than 1/10. What is the fewest possible number of students in her class?

(A) 19
(B) 20
(C) 21
(D) 71
(E) 73

(A) 19
(B) 20
(C) 21
(D) 71
(E) 73

chetan2u · Accepted Answer

Hi..
 The proper method would be 
 let n be the number, so n-1 are ODD prime.. 
1) choosing any 2 out of n is nC2..
2) ways in which sum is not even..
 one has to be 2 and other any from n-1.. so n-1C1..
 
Probability = \frac{n-1C1}{nC2}= \frac{(n-1)!}{(n-1-1)!1!}*\frac{(n-2)!2!}{n!}=(n-1)*\frac{2}{n(n-1)}=\frac{2}{n} ..

so from Q  \frac{2}{n}<\frac{1}{10}  or n>20..
so ans is n=21
C

kumarparitosh123 · Answer

The first 23 prime numbers 2,3,5,7,9,....67, 71 and 73. Selecting 2 out of 23is 23C2 means 253
And except 2 if we add any number to the other we will stand ven number only. So total outcomes are (2,3), (2,5), (2,7)......(2,73), leading to total of 20 outcomes.

So probability is 20/253 which is less than 1/10.

But if we select 19 or 20 numbers then the probability turns out to be ≥1/10 which is not acceptable as per te question stem.
So my Answer is C.

Sent from my Lenovo TAB S8-50LC using  GMAT Club Forum mobile app

Luckisnoexcuse · Answer

Let the assignment be for 2 students then 2, 3 probability of even sum (odd+odd is required) is 0 (2 is compulsory, rest all prime numbers are odd)
Let the assignment be for 3 students then 2, 3, 5 probability of even sum is 1/3
Let the assignment be for 4 students then 2,3,5,7 probability of even sum is 1/2
Let the assignment be for 5 students then 2,3,5,7,11 probability of even sum is 6/10
Now we know 
(n-1)C2/nC2 = Probability of sum even
for n = 19 
18C2/19C2 = 9*17/19*9 = 17/19 not even will be 2/19
for n=20
19*9/19*10 = 9/10 not even will be 1/10
for n=21
10*19/21*10 = 19/21 not even will be 2/21
for n=71
35*69/35*71 = 69/71 not even will be 2/71
for n = 73
72C2/73C2 = 71/73 not even will be 2/73

Fewest will be 21 
Option C

Ved22 · Answer

why to worry with such formulas

1. The prob of random 2 selection has to be odd , which means 2 will be a number in same. The number od distinct numbers == number of students
2. Next order matters, at random pick even+odd = odd or pick odd+even = odd. 
3. So, if the prob of one round will be considering option B . 1/20+19/19 == 1/20 , now considering step 02 the actual prob becomes 2/20 == 1/10.
4. The only next option that seems to be less than 1/10 is option C i.e 1/21

007saisurya · Answer

its an easy question 
The only way sum of two primes can be odd is when 2 is one of the primes. Every other prime is even(odd + odd)
assume x students are there in class
ways of choosing - 2 and another student = xC1
ways of choosing any two students = x+1C2
probability is xC1/x+1C2 <1/10
Solve - X>19 
so x>=20 and hence number of students >=21(including the 2)

shawrochis · Answer

Hello!,
So, if we calculate the probability as 2! * (1/x) * (x-1)/ (x-1) = 2/ x < 1/10 which leads to x> 20. 
Is this a correct way?

bumpbot · Answer

Automated notice from GMAT Club BumpBot:

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

This post was generated automatically.

A math teacher assigns a distinct prime number, starting with the lowe

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 math teacher assigns a distinct prime number, starting with the lowe

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