A random maths puzzle

[kamil9876]

Hi. Let a,b,c,d be the number of 150,100,75,60 wat bulbs used respsectively.

We have:

150a+100b+75c+60d=1000

And now we will keep reducing this equation:

30a+20b+15c+12d=200

From here on we see that c must be even and d must be a multiple of 5, hence let c=2k, d=5x:

30a+20b+30k+60x=200

3a+2b+3k+6x=20

Now we can see that b cannot be a multiple of 3, hence b is of the form b=3m+1 or b=3m+2 (mutually exclusive cases). By plugging in the second case we get:

3a+2(3m+2)+3k+6x=20
3a+6m+3k+6x=16
3(a+2m+k+2x)=16 which implies 16 has 3 as a factor, a contradiction. Hence only the first case is possible:

3a+2(3m+1)+3k+6x=20
3a+6m+3k+6x=18
a+2n+k+2x=6

And now find how many different solutions there are to that equation. Which is done manually or by computer or maybe generating functions (dunno what is quickest so far).

Now to finish it off:

(a+k)+2(n+x)=6

We have four cases:

a+k=0, 2(n+x)=6. 1 way to get a+k=0, 4 ways to get n+x=3

a+k=2, 2(n+x)=4. 3 ways to get a+k=2, 3 ways to get n+x=2

a+k=4, 2(n+x)=2. 5 ways to get a+k=4, 2 ways to get n+x=1

a+k=6, 2(n+x)=0. 7 ways to get a+k=6, 1 way to get n+x=0

Hence the final answer

[Momo.05]

o.o

[kamil9876]

ahh yes sorry, 75/5=15. lol ill edit now

wow, noticing that blunder led to a better simplification, check out new edited post

wow much better, yeah now generating functions work better...

Could someone tell me the basics of how you generate functions?

You don't really "generate functions". It's more like a function whose only purpose is to encode information in the coefficients etc. For example:



encodes information about . Now you can do stuff like plug in x=-1 to get a very useful identity:



Or you can differentiate it and set x=1 to get another fancy identity:



So generating functions allow you to do combinatorics without actually doing much thinking, but mostly algebraic manipulation. Of course there are more interesting examples of how generating functions can be used to derive all sorts of deep facts with minimal effort. A favourite of mine that converted me to and convinced me of the power of generatingfunctionology is the application of it to Partitions of an integer.

[Ahmad]

Generating functions :smitten:

[kamil9876]

gtfo! Elementary/Erdosian/pigeonhole principle type arguments FTW

[the.watchman]

gtfo! Elementary/Erdosian/pigeonhole principle type arguments FTW

LOL, everyone loves pigeonhole principle!!!

[TrueTears]

Generating functions :smitten:

mmm and you passed the love to me