How many employees are in company C? 1) There are 120 ways to form a

we can solve this DS using group based combination where order is not important

#1There are 120 ways to form a team of 3 out of all the employees in company C
(mn)!/(n!)^m*m! number of ways to divide into groups where order is not imp ...
(3m)!/(3!)^m* m! = 120
OR we can say nc3*n-3c3*n-6c3 = 720
sufficient
#2
All the employees in company C can be divided into two teams of equal employees in 126 ways
(2m)!/(2!)^m* m! =126

sufficient
OPTION D

1. nC3=120
2. nC2=126

D

Hi Archit...... Could you please explain again.... Having difficulty understanding the concept.

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Bunuel can you please help with this one


As per my understanding, state 1 gives nC3 = 120 and that can be solved to find out n=10. But how can we proceed with equation 2, I don't think that means nC2, as per my understanding that should be nCn/2

This is exactly what I thought too, is (2) not telling us that we need to do "nC(n/2)"?

Target question: How many employees are in company C?

Statement 1: There are 120 ways to form a team of 3 out of all the employees in company C
If we let n = the number of employees, then we can write: nC3 = 120
After testing a few numbers we can see that 10C3 = 120, which means n = 10, which means there are 10 employees in company C.
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: All the employees in company C can be divided into two teams of equal employees in 126 ways
This tells us that n (number of employees) must be an EVEN number.
It also tells us that [nC(n/2)]/2 = 126, which means nC(n/2) = 252
Let's test a few EVEN values of n:
If n = 2, then our equation becomes 2C1 = 252. Since 2C1 = 2, we know that n ≠ 2
If n = 4, then our equation becomes 4C2 = 252. Since 4C2 = 6, we know that n ≠ 4
If n = 6, then our equation becomes 6C3 = 252. Since 6C2 = 20, we know that n ≠ 8

Important: At this point we can see that, with each different even value of n, we get a different value for nC(n/2).
In other words, n can have exactly one value, statement 2 is sufficient to find the value of n.

Aside: If we keep going, we find that 10C5 = 252 which means n = 10, which means there are 10 employees in company C.
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Answer: D

ASIDE: Let me explain my equation [nC(n/2)]/2 = 126
n = the number of employees
n/2 = HALF the employees to be placed on one team (say Team A)
So, nC(n/2) = the number of ways to place half of the employees on Team A (which means the other half goes on Team B).
However, in this scenario we don't have a Team A and Team B; we just have two teams.
This means the expression nC(n/2) is DOUBLE COUNTING of a total number of outcomes, which means we must divide it by 2.