Finite Sequences

Probability Level 3

How many finite sequences $x_1,x_2,\cdots ,x_m$ are there such that each $x_i = 1$ or $2$ , and $\displaystyle \sum_{i=1}^{m} x_i = 10$ ?

92 91 89 120

1 solution

Jaydee Lucero
Jun 26, 2017

There are six possible cases:

For $\{2,2,2,2,2\}$ , there is only $1$ possible sequence.

For $\{1,1,2,2,2,2\}$ , there are $\displaystyle{\binom{6}{4}}=15$ possible sequences.

For $\{1,1,1,1,2,2,2\}$ , there are $\displaystyle{\binom{7}{3}}=35$ possible sequences.

For $\{1,1,1,1,1,1,2,2\}$ , there are $\displaystyle{\binom{8}{2}}=28$ possible sequences.

For $\{1,1,1,1,1,1,1,1,2\}$ , there are $\displaystyle{\binom{9}{1}}=9$ possible sequences.

And, for $\{1,1,1,1,1,1,1,1,1,1\}$ , there is only $1$ possible sequence.

Thus, all in all, there are $1+15+35+28+9+1=\boxed{89}$ possible sequences.