Red, orange, yellow, green and blue balls

Probability Level 4

You have a bag with 1 red ball, 2 orange balls, 3 yellow balls, 4 green balls, and 5 blue balls.

You pick three out (without replacement) at random.

The probability each is a different color is $\dfrac ab$ , where $a$ and $b$ are coprime positive integers.

What is $a+b$ ?

The answer is 136.

1 solution

Geoff Pilling
Sep 5, 2016

The total number of ways you can pick $3$ different colors is $\sum_{i,j,k \text{ different colors}}{}n_in_jn_k = 255$ The total number of ways you can choose $3$ balls is $\binom{1+2+3+4+5}{3} = 455$

So, the probability is $225/455$ or $45/91$ .

$45+91 = \boxed{136}$

How did you get 225 and 455?

Pi Han Goh - 4 years, 9 months ago

I have added it to the solution... I am realizing more and more that you are the "incomplete solution" police, and keep folks on their toes and honest! ;^)

Geoff Pilling - 4 years, 9 months ago

$225 = \sum_{i,j,k \text{ different}}{}n_in_jn_k$

$455 = \binom{1+2+3+4+5}{3}$

Geoff Pilling - 4 years, 9 months ago

In this question are we only focusing on the combinations of the colors? For example is r,y,o considered different from o,r,y?

Ashish Sacheti - 4 years, 9 months ago

All we care about is whether they are all different or not... The probability that they are all different.

Geoff Pilling - 4 years, 9 months ago

Red, orange, yellow, green and blue balls

The answer is 136.

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