# Question About HW 1, Problem 5

Is the expression on the right-hand side of the equation correct? Here’s why I’m asking: For $k = 2$, the denominator of this expression is $1 - 2x + x^2$, or $(1 - x)^2$, so the fraction reduces to $1/(1 - x)$. But this implies that $c_2 (n) = 1$ for all $n$, which is not true.

David

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### 3 comments

1. Maybe it is supposed to say that every part is strictly less than $k$, not less than or equal to. Because then that would be true since it’s just the one composition of all 1’s, right?

Hannah

2. Hannah is right; $c_k(n)$ should be the number of compositions of $n$ such that all parts are *strictly less* than $k$. Hannah showed us how to check the case $k=2$, and I encourage you to also think about the case $k=3$.
Thanks for the heads up, David.

3. Frits Scholer

EC1 problem 26 uses (k+1) so that would be the correct exponent in the formula. Love your lectures after buying the book (spent 2 days on hw1 problem 3 and found alternative solution based on induction) Very stimulating problems.
Frits, Amsterdam