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# Lizzie has a strange deck of number cards.

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Lizzie has a strange deck of number cards that contains cards labeled 2 through 9 in three different colors (red, blue, and green).

(a) In how many ways can she draw one card from each color so that the sum of all the cards is 10?

(b) In how many ways can she draw one card from each color so that the sum of all the cards is 20?

Colors matter!

Oct 20, 2018

#1
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You have 9 - 2 + 1 = 8 cards with red, blue and green colors. How are the 3 colors distributed among the 8 cards?

Or, should your cards be labeled 1 through 9?.

Oct 20, 2018
#2
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I think she has a card numbered 2,...,9 for each color (8 red cards numbered from 2-9, 8 blue cards numbere from 2-9 and 8 green cards numbered from 2-9)

Guest Oct 20, 2018
#3
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Here is a hint for anybody who wants to try it:

You can only make the sum of 10 in four different ways as follows:

6 + 2 + 2 = 10 - permutations =3! / 2! =3
5 + 3 + 2 = 10 - ....................  =3!      = 6
4 + 4 + 2 = 10 - ....................  =3! / 2! =3
4 + 3 + 3 = 10 - ....................  =3! / 2! =3

And you can only make the sum of 20 in eight different ways as follows:

9 + 9 + 2 = 20 - permutations = 3! / 2! =3
9 + 8 + 3 = 20 - ..................... = 3!       =6
9 + 7 + 4 = 20 - ..................... = 3!       =6
9 + 6 + 5 = 20 - ..................... = 3!       =6
8 + 8 + 4 = 20 - ..................... = 3! / 2! =3
8 + 7 + 5 = 20 - ..................... = 3!       =6
8 + 6 + 6 = 20 - ..................... =3! / 2!  =3
7 + 7 + 6 = 20 - ..................... =3! / 2!  =3

That is as far as I go!!!.

Oct 20, 2018
#4
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that's I thought I should do but I'm not sure if there's a better way

SaltyGrandma  Oct 20, 2018
#5
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I think its a) 90 and b) 216

Please correct me and show me how to do this problem if I'm wrong

Oct 20, 2018
#6
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I'm thinking that each 3-number combination for making up a 10 should be:

3^3 x 4 =108 ways.

And each 3-number combination for making up a 20 should be:

3^3 x 8 =216 ways.

Oct 20, 2018
#7
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This was what I was thinking for (a):

There are 3!*(3+6+3+3) ways so it's 90.

(3! is for the colors, 3+6+3+3 is the number of ways the sum is 10)

So how am I supposed to do it then?

Oct 20, 2018
edited by SaltyGrandma  Oct 20, 2018
#8
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The way I came up with 108 and 216 is as follows:

Take the first combination to make up a 10: 6 + 2 + 2 =10. But the first 6 comes in 3 colors: red, blue, green.

The second 2 also comes in 3 colors: red, blue, green. The same thing goes for the third 2. So, we have a combination of 3 x 3 x 3 = 27 possible ways to make up 6 + 2 + 2 =10. And this should apply to the other 3 remaining combinations of 10. So, 27 x 4 =108 ways of making up a 10.

Exactly the same reasoning applies to making up the 20s, 3 x 3 x 3 x 8 =216 ways.

That is my best guess!.

Oct 20, 2018
#9
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This is a clear violation of the AoPS honor code.

The AoPS honor code clearly states that you may not, in any case use outside resources to figure out solutions to problems.