well originally the question was find the number of ways to make change for 1 dollar using up to 9 pennies and any number of nickels and dimes.
I just figured it was implied, my bad.
I just reduced it about as far as I can, but I dont know how to get the number of coefficients easily.
not sure if theres an easy way to do it, but my professor did it in his head lol.
yea, I originally had the problem like this..
I then reduced the first expression to :
left the second expression untouched, and reduced the third to, .
I actually have the problem worked out, but the last line where the professor got the answer left me blank, because I cant see how he got 21 so easily.
He explained it, and told us the logic to what the answer was, undeniably I couldnt really write down what he was saying, but what he said made sense.
I just cant figure it out.
It was also a review problem for my test tomorrow but these binomial coefficient problems are kicking my ass.
Can you help me with this problem then..
I need the coefficient of
Let: if 5 doesn't divide n, and if 5 does divide n
The coefficient of stays untouched if we change for
We have
Thus:
But: is the number of multiples of 5 between 0 and n, thus we have where is the floor function
So the coefficient of turns out to be
Stated in another way: For each , you have only one element among such that the product of the pair gives . It follows then that the coefficient of is actually 21, which is the numbers of terms of the factor