2011 Invitational A (Probability Questions)

[Jeremy2012x]

I did pretty well, but I'd still like to figure out the ones I just had no clue on (especially probability haha)

20) The directrix of the conic given by the equation [unparseable or potentially dangerous latex formula]

32) Triangle PQR is similar to Triangle XYZ. [unparseable or potentially dangerous latex formula] and [unparseable or potentially dangerous latex formula] Find PR + YZ

36) Three letters are randomly chosen from the word EIGHT. What are the odds that the set of three letters randomly chosen consist of at least 2 consonants?

54) The Sweet Tooth candy store has four different types of gum balls, a sweet ball, a sour ball, a hot ball, and a red hot ball. They package them in sets of three. How many ways can the store package the gum balls?

55) Coach Wenzen recruits 5 physics students, 4 chemistry students, and 3 biology students for his science club. How many different 6-member science teams can he form if each team consists of 3 physics students, 2 chemistry students, and 1 biology?
For this one, I realize the answer is 180 (as it is the product of the combination for each subject), however, can someone explain to me why this so? why not add each combination?

60) Ima Sportnutt has 5 baseball cards, 3 pitchers and 2 catchers. She has 6 football cards, 4 quarterbacks and 2 centers. Ima randomly selects two cards without replacement. What is the probability that both cards are quarterbacks? (nearest per cent)

[nsguy1350]

I believe that the standard form (or a form similar to the standard form) for a parabola is [unparseable or potentially dangerous latex formula]. For an up-down parabola pointing downwards, the directrix is given by [unparseable or potentially dangerous latex formula], I believe.

Set up similar triangles and ratios. That should be simple enough.

There are [unparseable or potentially dangerous latex formula] combinations of letters. Writing them out, you get that 3 do not have 2 consonants, and 2 vowels instead. This gives [unparseable or potentially dangerous latex formula] odds.

54. Since you can pick each thing multiple times, I believe you have [unparseable or potentially dangerous latex formula] choices. I think this is a "ball-and-pins" type problem, so you do [unparseable or potentially dangerous latex formula], using standard combination notation.

55. Good thing you know how to do it.
Lets say you have x combinations for the first one, and y combinations for the second one.
For each combo in the x, you have all of the y combinations that can go with it.
This means you have to multiply.

For example, lets say I can pick a number from 1-3 and then a number from 4-5.
I need to multiply the number of combos of the first one (3) by the second (2) to get the total amount of combinations.
Adding them does not work.

[Salvadoramerican]

the last one their are [unparseable or potentially dangerous latex formula] total combinations then for football quarterback cards their are 4 so [unparseable or potentially dangerous latex formula] combos so divide the quarterbacks combos by the total 2 combos, hey I did it my first Latex also I see your learning nsguy that good

[nsguy1350]

Thanks! Oh, I seem to have forgotten to do the last problem. Salvadoramerican covered it up for me though, so it's fine.