$\sum ^{m}_{i=1} x_i =10$, where $x_i=1\; or \;2$
All Ten are $1's$ = $1$ choice
One is $2$, and eight $1's$ = $^9C_1=9$ choices
Two $2's$, and six $1's$ = $^8C_2 =28$
Three $2's$, and four $1's$ = $^7C_3 =35$
Four $2's$, and two $1's$ = $^6C_4 =15$
All Five are $2's$ =$1$
Total finite sequence $= 1+9+28+35+15+1=89$