Stuck on a hard math question. Math people this is for you!?

Question

8, 7, 6, 3, 2 are different ways you can score points in a football game. If the bears scored 39 points in a game, how many combinations can you come up with to get a total of 39 points?

Answer 1

a lot!
these are all the ways i came up with:
8,8,8,8,7
8,8,8,8,3,2,2
8,8,8,7,6,2
8,8,8,7,3,3,2
8,8,8,6,6,3
8,8,8,6,3,3,3
8,8,8,6,3,2,2,2,
8,8,8,3,3,3,3,3
8,8,8,3,3,3,2,2,2
8,8,8,3,2,2,2,2,2,2
8,8,7,7,7,2
8,8,7,7,6,3
8,8,7,7,3,3,3
8,8,7,7,3,2,2,2
8,8,7,6,6,2,2
8,8,7,6,3,3,2,2
8,8,7,6,2,2,2,2,2
8,8,7,3,3,3,3,2,2
8,8,7,3,3,2,2,2,2,2
8,8,7,2,2,2,2,2,2,2,2
8,8,6,6,6,3,2
8,8,6,6,3,3,3,2
8,8,6,6,3,2,2,2,2
8,8,6,3,3,3,3,3,2
8,8,6,3,3,3,2,2,2,2
8,8,6,3,2,2,2,2,2,2,2
8,8,3,3,3,3,3,3,3,2
8,8,3,3,3,3,3,2,2,2,2
8,8,3,3,3,2,2,2,2,2,2,2
8,8,3,2,2,2,2,2,2,2,2,2,2
8,7,7,7,7,3
8,7,7,7,6,2,2
8,7,7,7,3,3,2,2
8,7,7,7,2,2,2,2,2
8,7,7,6,6,3,2
8,7,7,6,3,3,3,2
8,7,7,6,3,2,2,2,2
8,7,7,3,3,3,3,3,2
8,7,7,3,3,3,2,2,2,2
8,7,7,3,2,2,2,2,2,2,2
8,7,6,6,6,6
8,7,6,6,6,3,3
8,7,6,6,6,2,2,2
8,7,6,6,3,3,3,3
8,7,6,6,3,3,2,2,2,
8,7,6,6,2,2,2,2,2,2
8,7,6,3,3,3,3,3,3
8,7,6,3,3,3,3,2,2,2
8,7,6,3,3,2,2,2,2,2,2
8,7,6,2,2,2,2,2,2,2,2,2
8,7,3,3,3,3,3,3,3,3
8,7,3,3,3,3,3,3,2,2,2
8,7,3,3,3,3,2,2,2,2,2,2
8,7,3,3,2,2,2,2,2,2,2,2,2
8,7,2,2,2,2,2,2,2,2,2,2,2,2
8,6,6,6,6,3,2,2
8,6,6,6,3,3,3,2,2
8,6,6,6,3,2,2,2,2,2
8,6,6,3,3,3,3,3,2,2
8,6,6,3,3,3,2,2,2,2,2
8,6,6,3,2,2,2,2,2,2,2,2
8,6,3,3,3,3,3,3,3,2,2
8,6,3,3,3,3,3,2,2,2,2,2
8,6,3,3,3,2,2,2,2,2,2,2,2
8,6,3,2,2,2,2,2,2,2,2,2,2,2
8,3,3,3,3,3,3,3,3,3,2,2
8,3,3,3,3,3,3,3,2,2,2,2,2
8,3,3,3,3,3,2,2,2,2,2,2,2,2
8,3,3,3,2,2,2,2,2,2,2,2,2,2,2
8,3,2,2,2,2,2,2,2,2,2,2,2,2,2,2
7,7,7,7,7,2,2
7,7,7,7,6,3,2
7,7,7,7,3,3,3,2
7,7,7,7,3,2,2,2,2
7,7,7,6,6,6
7,7,7,6,6,3,3
7,7,7,6,6,2,2,2
7,7,7,6,3,3,3,3
7,7,7,6,3,3,2,2,2
7,7,7,6,2,2,2,2,2,2
7,7,7,3,3,3,3,3,3
7,7,7,3,3,3,3,2,2,2
7,7,7,3,3,2,2,2,2,2,2
7,7,7,2,2,2,2,2,2,2,2,2
7,7,6,6,6,3,2,2
7,7,6,6,3,3,3,2,2
7,7,6,6,3,2,2,2,2,2
7,7,6,3,3,3,3,3,2,2
7,7,6,3,3,3,2,2,2,2,2
7,7,6,3,2,2,2,2,2,2,2,2
7,7,3,3,3,3,3,3,3,2,2
7,7,3,3,3,3,3,2,2,2,2,2
7,7,3,3,3,2,2,2,2,2,2,2,2
7,7,3,2,2,2,2,2,2,2,2,2,2,2
7,6,6,6,6,6,2
7,6,6,6,6,3,3,2
7,6,6,6,6,2,2,2,2
7,6,6,6,3,3,3,3,2
7,6,6,6,3,3,2,2,2,2
7,6,6,6,2,2,2,2,2,2,2
7,6,6,3,3,3,3,3,3,2
7,6,6,3,3,3,3,2,2,2,2
7,6,6,3,3,2,2,2,2,2,2,2
7,6,6,2,2,2,2,2,2,2,2,2,2
7,6,3,3,3,3,3,3,3,3,2
7,6,3,3,3,3,3,3,2,2,2,2
7,6,3,3,3,3,2,2,2,2,2,2,2
7,6,3,3,2,2,2,2,2,2,2,2,2,2
7,6,2,2,2,2,2,2,2,2,2,2,2,2,2
7,3,3,3,3,3,3,3,3,3,3,2
7,3,3,3,3,3,3,3,3,2,2,2,2
7,3,3,3,3,3,3,2,2,2,2,2,2,2
7,3,3,3,3,2,2,2,2,2,2,2,2,2,2
7,3,3,2,2,2,2,2,2,2,2,2,2,2,2,2
7,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2
6,6,6,6,6,6,3
6,6,6,6,6,3,3,3
6,6,6,6,6,3,2,2,2
6,6,6,6,3,3,3,3,3
6,6,6,6,3,3,3,2,2,2
6,6,6,6,3,2,2,2,2,2,2
6,6,6,3,3,3,3,3,3,3
6,6,6,3,3,3,3,3,2,2,2
6,6,6,3,3,3,2,2,2,2,2,2
6,6,6,3,2,2,2,2,2,2,2,2,2
6,6,3,3,3,3,3,3,3,3,3
6,6,3,3,3,3,3,3,3,2,2,2
6,6,3,3,3,3,3,2,2,2,2,2,2
6,6,3,3,3,2,2,2,2,2,2,2,2,2
6,6,3,2,2,2,2,2,2,2,2,2,2,2,2
6,3,3,3,3,3,3,3,3,3,3,3
6,3,3,3,3,3,3,3,3,3,2,2,2
6,3,3,3,3,3,3,3,2,2,2,2,2,2
6,3,3,3,3,3,2,2,2,2,2,2,2,2,2
6,3,3,3,2,2,2,2,2,2,2,2,2,2,2,2
6,3,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2
3,3,3,3,3,3,3,3,3,3,3,3,3
3,3,3,3,3,3,3,3,3,3,3,2,2,2
3,3,3,3,3,3,3,3,3,2,2,2,2,2,2
3,3,3,3,3,3,3,2,2,2,2,2,2,2,2,2
3,3,3,3,3,2,2,2,2,2,2,2,2,2,2,2,2
3,3,3,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2
3,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2
that comes to a total of 143 combinations i believe....that took forever
hope this helps

Answer 2

The answer is indeed 143 ways as Kevin pointed out.
But here is the mathematical proof:

For students of combinitorics, the problem is equivalent to finding the number of partitions of the number 39 with summands restricted to the values 8,7,6,3 and 2. This can be solved using the method of generating polynomials.
Consider the following polynomials:

P1 = 1 + x^2 + x^4 + x^6 + x^8 + ...
P2 = 1 + x^3 + x^6 + x^9 + x^12 + ...
P3 = 1 + x^6 + x^12 + x^18 + x^24 + ...
P4 = 1 + x^7 + x^14 + x^21 + x^28 + ...
P5 = 1 + x^8 + x^16 + x^24 + x^32 + ...

The answer to the question is then the coefficient of x^39 in the expansion of P1 * P2 * P3 * P4 * P5

It may take a while, but if you are careful and bother to expand this product (or use Mathematica, Maxima, or some other program) you will find that it is:

... + 143*x^39 + 139*x^38 + 121*x^37 + 119*x^36 + 103*x^35 + 99*x^34 + 87*x^33 + 84*x^32 + 71*x^31 + 71*x^30 + 59*x^29 + 57*x^28 + 49*x^27 + 47*x^26 + 38*x^25 + 39*x^24 + 31*x^23 + 30*x^22 + 25*x^21 + 24*x^20 + 18*x^19 + 19*x^18 + 14*x^17 + 14*x^16 + 11*x^15 + 11*x^14 + 7*x^13 + 8*x^12 + 5*x^11 + 5*x^10 + 4*x^9 + 4*x^8 + 2*x^7 + 3*x^6 + x^5 + x^4 + x^3 + x^2 + 1

Each other coefficient in the product above represents the number of ways to get ANY number of (fewer than 39) points in a football game, the number of points being the exponent in the term. i.e., there are 18 ways to score 19 points, 71 ways to score 31 points, etc.

Answer 3

Well, we know you can't score less than 0 8s nor more than 4 8s, so that means the Bears can score 0, 1, 2, 3, or 4 8s.

Let's use similar logic for the remaining possible scores:

8: 0-4
7: 0-5
6: 0-6
3: 0-13
2: 0-19

This gives us a total of 4*5*6*13*19 = 29640 combinations. Now we have to eliminate combinations because not all of these will yield a score of 39 obviously!

This could get messy...

one 8
-------

we need to score 31 with the remaining combinations... which means:

7: 0-4
6: 0-5
3: 0-10
2: 0-15

Man, this is gonna be ugly...

8 8 8 8 7
8 8 8 8 3 2 2
8 8 8 7 3 3 2
8 8 8 7 6 2
8 8 8 7 2 2 2 2
8 8 8 7 6 2
8 8 8 3 2 2 2 2 2 2
8 8 8 3 3 3 2 2 2
8 8 8 3 3 3 3 3
8 8 7 7 7 2
8 8 7 7 3 3 3
8 8 7 7 3 2 2 2

ugh, I don't have the patience to do this... I need to figure out a formula :P

Answer 4

Alright..
This could be it!
I've instead of looking for all of the possiblilities for 39, I checked the numbers for 1,2,3,4,5,6,7,8,9,10,11. I 've discovered a pattern, the possibilities increase by 1 0 0 0 1 0 0 2, then the pattern repeats. Therfore, using the pattern i must come up with a mathematicla formula.
Screw that.
39 divisible by 8, 4 times with a remainder 7.
So, since the pattern moves up 4 every 8, 32 is the possible solutions, not including 2. So 34 is the number of possible soultions...
Screw that.
Call a math teacher or someone, cause i have no clue what the formula is!
I've research this for about an hour and still no results!?!?!?
Anyway, if 34 is the true answer I found it by pure luck but, you should still reward my efforts.

Answer 5

You can only get 4 8s in there (right, like 4 2-pt conversions would ever happen). The 7 pts left over could be a single 7-pt score (though if the Bears were on such a roll, I imagine they'd try for another 8 pts). Or it could be a field goal and two safeties. Anyhow, that's two possibilities, all there are to max out 8s. So, how about 3 8s? That leaves 15 pts unaccounted for. These could be blamed on a 7 pointer and 4 safeties, or a 7 pointer, two field goals and a safety. Or a missed pat plus 3 field goals, or a missed pat plus 3 safeties and a field goal

Dang, there must be a formula for something like this.

Why don't we just wait till Sunday and see what the Bears actually do score?

Answer 6

lets see...

39 = 8+8+8+8+7
but each 8 could be 6+2, or 3+3+2, or 2+2+2+2, so each 8 has 3 possibilities, so that makes another 3 possibilities for 8, or two 8's, or 3 8's, or all 4 8's, making 3*4 more possiblities...

Then there's things like... 6+6+6+6+6+6+3
but each 6 could be 2+2+2 or 3+3, so that's 2*6 more possibilities...

Or then there's the whole 3*13 one..
Maybe the 3*11+6, or the 3*9+2*6, or the...

So I'm going with, more than 28

If you realllly want to know, I think it's a matter of being stubborn enough to write it out. I don't know a good short cut to this one.

Answer 7

Its not hard.

Just fool around with a calculator with those numbers and keep track of how many times you come up with a combination.

Answer 8

13 feild goals,

Answer 9

ummmm....

Answer 10

maybe 195 ways..............lol