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How many finite sequences $x_{1}, x_{2},...,x_{m}$ are there such that each $x_{i}=1$ or $2$, and $\sum_{i=1}^{m} x_{i}=10$ ?

1. $89$
2. $91$
3. $92$
4. $120$
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$\sum ^{m}_{i=1} x_i =10$, 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 $= 1+9+28+35+15+1=89$

by Veteran (56.6k points)
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+1

1, 2, 3, 5, 8, 13, 21, 34,,55,89=FIBONACCI SERIES.DIRECTTLY

0
Sir , can u plz explain once again how u got 8C2 for the case when we have 2 2's and 6 1's .
0
we have 8 numbers/places ,
either place 2 2's, then rest are 1's by defaul, t 8C2
or place 6 1's rest are 2's by default , 8C6

+1 vote