organized counting: rolling dice, picking cards, T/F tests,

[xo_caroline_xo]

i have 4 question in my data management course which i dont clearly understand and i was wondering if anyone could help me, or even give me some advice in finishing my assignment...here are the questions;

2. In how many ways can you roll either a sum of 4 or a sum of 11 with a pair of dice?

3. In how many ways can you draw a 6 or a face card from a deck of 52 playing cards?

8. A standard die is rolled five times. How many different outcomes are possible?

10. In how many ways can a student answer a true-false test that has six questions. Explain your reasoning.

anyone who can help me would be appreciated, thanks!!

caroline

 

[galactus]

Re: organized counting

i have 4 question in my data management course which i dont clearly understand and i was wondering if anyone could help me, or even give me some advice in finishing my assignment...here are the questions;

2. In how many ways can you roll either a sum of 4 or a sum of 11 with a pair of dice?

There are \(\displaystyle 6^{2}\) possible outcomes with 2 dice. You can list them out and count or you could show off and use the generating functions

\(\displaystyle \L\\\left(\sum_{p=1}^{6}{x^{p}}\right)^{2}\)

Now, in the expanded expression, find the coefficients which go with the exponents 4 and 11.

We find \(\displaystyle x^{12}+2x^{11}+3x^{10}+4x^{9}+5x^{8}+6x^{7}+5x^{6}+4x^{5}+3x^{4}+2x^{3}+x^{2}\)

Pretty cool, huh?.

3. In how many ways can you draw a 6 or a face card from a deck of 52 playing cards?

There are 4 different 6's and 12 face cards. Any thoughts?.

8. A standard die is rolled five times. How many different outcomes are possible?

Well, if \(\displaystyle 6^{2}\) outcomes are possible with 2 dice, how many are possible with 5?.

10. In how many ways can a student answer a true-false test that has six questions. Explain your reasoning.

Same principle as the dice problem.

anyone who can help me would be appreciated, thanks!!

caroline

 

[pka]

Re: organized counting

3. In how many ways can you draw a 6 or a face card from a deck of 52 playing cards?
There are 4 different 6's and 4 face cards. Any thoughts?.

One minor correction: of course there are 12 face cards, 3 in each suite.

 

[galactus]

DUH, of course there are. JKQ-4 suits. That makes 12 face cards.

I will change my post. Thanks, pka. Major brain fart.

 

[xo_caroline_xo]

Re: organized counting

\(\displaystyle \L\\\left(\sum_{p=1}^{6}{x^{p}}\right)^{2}\)


i dont understand that formula since my teacher hasnt even taught us that, could you please explain that formula to me?....thanks!

caroline

 

[galactus]

Your teacher may not teach it. It's from more advanced classes in combinatorics.

From the generating function you can see the answers are 3(for the number of sums of 4) and 2(the number of sums of 11).

Your best bet would probably be to count them up.

For 4: 2 and 2; 3 and 1; 1 and 3

For 11: 6 and 5; 5 and 6

 

[xo_caroline_xo]

i guess i understand it more a bit, so i think i'll just use the formula you gave me and hopefully try to explain to my teacher how i/you did it and see what she has to say...but anyways thanks sooooooooooo much!!

caroline