how many ways are there to make change for quarter, using only dimes, nickels and penis?

Answer 1

First of all, i assume u meant pennies. :))
Now the different sets can be
Change made using only 1s, i'll represent this with C(1) = 1
Similarly C(5) = 1 and C(10) = 0
C(1,5) = 4, i.e. 25 can be split up as 1n + 20p or 2n + 15p or 3n + 10p or 4n + 5p.
Similarly C(1, 10) = 2
C(5, 10) = 2
C(1, 5, 10) = 2
Thus the total ways is 1+1+4+2+2+2 = 12 ways

Answer 2

The way to answer problems like this is to find a systematic way to count them, and just work through it.

For instance first start off with just penis, obviously 25 make a quarter, so that is one way. Then add a nickel (but no dimes yet) which means you only need 20 penis for the second way. Keep adding nickels, until you have exhausted all the ways to do it with just nickels and penis. Then repeat the whole thing including exactly one dime, then two etc.