How many balls to draw to ensure balls of same color?

Question

We have a bag which contains $5$ red balls, $8$ blue balls, $10$ white balls, $12$ green balls and $7$ yellow balls. How many balls should we pick to ensure

• a) at least 4 balls have the same color?

• b) at least 6 balls have the same color?

• c) at least 7 balls have the same color?

• d) at least 9 balls have the same color?

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I think I solved the a) part. If we pick a ball every time till we reach $4$ balls with the same color it takes us $15+1$ balls.

But I don't understand the other parts. Can you guys help me? It will mean A LOT to me. Thanks

Answer 1

We look at the worst possible case, and draw one more ball.

1. Worst possible case was drawing $3$ balls of each of five colors first, before reaching to fourth ball of some color. Hence $3\cdot5+1$
2. Worst case includes drawing all 5 of red balls, then five each of other colors. Hence $5\cdot5+1$
3. $\, \ldots$

Similarly, can you proceed?