How to find the least number of extractions to make the word WAAY-TV?

Question

The problem is as follows:

A local tv station is doing a small lottery among their employees to celebrate the anniversary of the news department.

The owner has set five carton boxes which a set of colored plastic cards which have printed the letters WAAYTV (the callsign of his station).

The each box is put one next to another making a row, with the letter W positioned in the left and V on the right and has a label outside indicating its contents and these are:

Box W: 16 red cards and 15 blue cards

Box A: 16 red cards and 14 blue cards

Box Y: 16 red cards and 14 blue cards

Box T: 14 red cards and 15 blue cards

Box V: 14 red cards and 15 blue cards

How many of the cards need to be drawn one by one without looking in order to be sure to make the callsign of the station such that all the cards share the same color?

It is not specifically mentioned in the problem but I believe looking at similar problems from the same author. The intended meaning is that in this context it is drawing with no replacement.

I'm not sure exactly how to tackle this problem. But it seems to my that in order to guarantee to get all the letters sharing the same color has to do assuming the least likely to happen scenario. In other words looking for an option which assumes you get what you are not looking for and after exhausting these options you can be sure to get what you want.

Given this preamble, I'm assuming that, the contestant, stars taking out cards from the left.

If a card is taken out from the box there, we don't know if its either red or blue it could be any. So in order to compensate for this unknown we have to be sure to extract all the rest which aren't known.

I guess 14+15 from V and from T, and 16+14 from Y and 16 from A + 2 additional which ensure that those have the same color.

Adding all these would be

$1+14+15+14+15+16+14+16+2=107$

But is thia a correct assumption?. I don't know if I'm really doing it right, thus an answer which would help me the most is one which could explain what sort of logic orderly should be followed here to solve this. Since I'm slow at catching up the clues, I require that the answer could be step by step.

Answer 1

Here's a strategy that uses just $64$:

Choose $2$ As: if they match, continue below; if not, choose a third A. You are guaranteed to have two As of the same color. (Think about it.)

Now choose 16 Ws, 16 Ys, 15Ts and 15 Vs, and you'll be guaranteed to have all the same color.

Answer 2

I'm assuming that the number of picks from each box must be specified in advance.

Choosing 17W, 17Y, 16T, 16V, and 3A yields at least 1 of each color for WYTV and at least 2 of at least one color for A. The total is 69.

But a better solution is to choose 17W, 1Y, 16T, 16V, and 18A, which yields at least 1 of each color for WATV, and so either color of the Y card suffices. The total is 68, and this turns out to be optimal. An alternative optimal solution is 1W, 17Y, 16T, 16V, and 18A.