As a treat for her two crying children, a mother runs to the freezer

Whoa!! you make it sound so simple man. Kudos to you.

Probability of not getting the same flavor ->

Fouvarable -> Cherry - orange [ 2C1 * 3C1 [or simply 2*3 ] or Cherry - lemon [ 2*4] or orange - lemon [3*4]

Prob = (2*3 + 2*4 + 3*4 ) / 9C2 = 26/36 = 13/18

Answer -> D

-----------
Kudos if this helps

Official solution from Veritas Prep.

The mother will have to return to the freezer if each child receives a different flavor, meaning that she does not have to return to the freezer if they each receive the same flavor. And in probability, it's often more convenient to mathematically account for "matches" (or pairs) than it is for non-matches, so in this case it may be helpful to use the "outcomes when the ice pops match" and subtract from one than it is to go the other route.

If the goal, then, is to match, then there are three ways to do it:

Both cherry: 2/9 * 1/8

Both orange: 3/9 * 2/8

Both lemon-lime: 4/9 * 3/8

And since the denominator is the same in each case (you start with 9, then for the second draw there are 8 left, so the denominator is always 9 * 8), it's easier if you don't reduce the fractions before adding:

(2 * 1) + (3 * 2) + (4 * 3) all over the common denominator leads to:

(2 + 6 + 12)/72

20/72 = 10/36 = 5/18

But remember - 5/18 is the probability that she got what she wanted on her random draw, and we want the probability that she did not. That's the other 13/18 of outcomes, so the answer is D.

This is how I approached it:

Goal : Mom picked different flavors. -> Too difficult to picture all possible combinations
paraphrased goal = 1 - probability of mom picking up same flavors

Total possible ways to pick 2 out of 9 ice pops = 9C2 = 36

Case 1- total number of possibilities of picking up only cherry pops = 2C2 = 1
p(2 CP) = 1/36

Case-2 - total number of possibilities of picking up only orange pops = 3C2 = 3
p(2 OP) = 3/36

Case-3 total number of possibilities of picking up only lemon-lime pops = 4C2 = 6
p(2 LP) = 6/36

total prob. = (1+3+6)/36 = 10/36

goal = 1 - 10/36 = 26/36 = 13/18

Ans. is D

Can you please elaborate what I did wrong here:

P of not getting the same flavor = 1- P of getting the same flavor.

P of getting the same flavor = P of picking 2 C + P of picking 2 O + P of picking 2 L
P of picking 2 C out of 2 = 2*1 = 2
P of picking 2 O out of 3 = 3*2 = 6
P of picking 2 L out of 4 = 4*3 = 12
Sum of the 3 P is 20

So P of getting the same flavor is 20/36
So P of not getting the same flavor is 16/36 = 4/9

It's easier to subtract the probability of the mother choosing matching ice pops from 1:

P(2 cherry) = 2/9 * 1/8 = 2/72
P(2 orange) = 3/9 * 2/8 = 6/72
P(2 lemon lime) = 4/9 * 3/8 = 12/72
total = 20/72 = 5/18

1 - P(mother chose matching ice pops) = 1 - 5/18 = 13/18