Number of $1$ s in Compositions of $20$

Probability Level 3

A composition of a positive integer $n$ is an expression of $n$ as the sum of only 1s and 2s, where the order matters.

For example, $(1+1+1)$ , $(1+2)$ , and $(2+1)$ are all possible compositions of 3.
The number of total 1s in all three of these compositions is 5.

Find the total number of 1s in all possible compositions of 20.

1 solution

Kushal Bose
Dec 22, 2016

$1.x+2.y=20$ where $x$ is the number of $1's$ and $y$ is the number of $2's$

$1.x=20-2.y$ it can be concluded that $x$ should be even

Total ways= $100610$

@Muhammad Rasel Parvej A lovely sum.... . Although a tough calculation... Anyways it was a good one

Md Zuhair - 4 years, 5 months ago

I must appreciate the effort by @Kushal Bose . But there is better solution with less calculation. I'll post it later.

Muhammad Rasel Parvej - 4 years, 5 months ago

Ya there may be but while doing the sum I faced calculation difficulties too

Md Zuhair - 4 years, 5 months ago

@Md Zuhair – Exactly I have also faced many problems but I tried untill I get the result

Kushal Bose - 4 years, 5 months ago

@Kushal Bose – I used generating functions , but i was stuck with this sum $\displaystyle\sum_{r=0}^{10} (20-2r)\dbinom{20-r}{r}$ .

Sabhrant Sachan - 4 years, 5 months ago