Problem Solving

Kurt, a painter, has 9 jars of paint:
4 are yellow
2 are red
rest are brown

Kurt will combine 3 jars of paint into a new container to make a new color, which he will name accordingly to the following conditions:

Brun Y if the paint contains 2 jars of brown paint and no yellow
Brun X if the paint contains 3 jars of brown paint
Jaune X if the paint contains at least 2 jars of yellow
Jaune Y if the paint contains exactly 1 jar of yellow

What is the probability that the new color will be Jaune

a) 5/42
b) 37/42
c) 1/21
d) 4/9
e) 5/9

Source: https://www.beatthegmat.com/difficult-gm ... 8.html#116

I started solving this problem and was struck with a fundamental doubt. If there are n identical objects and if we are to make groups of r objects (r < n), how many groups (or combinations) are possible?

If the objects are not identical, then it is nCr = n! / ((n - r)! x r!). But what if the objects are identical, is it the same?

Take for example, the 4 yellow jars in the question.

Case 1: If they are non-identical i.e. Y1, Y2, Y3, Y4 then the number of ways of picking 3 jars is 4C3.

Case 2 (this is what I'm confused about): If the yellow jars are all identical, Y, Y, Y and Y. Any 3 that I pick are going to be the same YYY and I won't be able to differentiate which 3 these are. So is the number ways of picking 3 jars out of the 4 identical ones in the case 1?

I would really appreciate help from any of the experts.

santhoshsram wrote:Kurt, a painter, has 9 jars of paint:
4 are yellow
2 are red
rest are brown

Kurt will combine 3 jars of paint into a new container to make a new color, which he will name accordingly to the following conditions:

Brun Y if the paint contains 2 jars of brown paint and no yellow
Brun X if the paint contains 3 jars of brown paint
Jaune X if the paint contains at least 2 jars of yellow
Jaune Y if the paint contains exactly 1 jar of yellow

What is the probability that the new color will be Jaune?

a) 5/42
b) 37/42
c) 1/21
d) 4/9
e) 5/9

P(Jaune) = 1 - P(not Jaune).
The new color will not be Jaune if it omits yellow.

P(not Jaune):
P(first jar is not yellow) = 5/9. (9 jars, 5 of them not yellow)
P(second jar is not yellow) = 4/8. (8 jars left, 4 of them not yellow)
P(third jar is not yellow) = 3/7. (7 jars left, 3 of them not yellow)
To determine the probability that all of these events happen together, we multiply the fractions:
5/9 * 4/8 * 3/7 = 5/42.

P(Jaune):
1 - 5/42 = 37/42.

The correct answer is B.

i think, Santosh is asking about whether with no difference between the jars of the same color the solution is different. The general assumption here is that the jars of the same color are different/unique. Otherwise you have different approach,

to select 3 jars of the same color --> 3 ways (all brown, red or yellow)
to select 3 jars of different colors --> 1 way (yellow-brown-red in different combination is the same)
to select 2 colors the same and 1 is different --> 6 ways (brown-brown-x, yellow-yellow-x, red-red-x)
:: total 10 ways

to select 'Jaune X if the paint contains at least 2 jars of yellow' --> 3 ways
to select 'Jaune Y if the paint contains exactly 1 jar of yellow' --> 3 ways
:: total 6 ways

Thus, it makes 6/10 or 3/5

the result we obtained is not among the listed choices, and I keep my assumption here that the jars of the same color are different/unique.