Ones, twos and threes
How many ordered sequences of ones, twos and threes sum up to 10?
Examples: (3,2,2,2,1), (3,3,3,1),(3,3,1,3), ....
The answer is 274.
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1 solution
Moderator note:
We are essentially looking for the coefficient of x^[10} in the power series expansion of . Unfortunately, there isn't a simple way of evaluating this.
There are 14 cases where 1s, 2s and 3s add up to 10 and each case can be permutated a certain number of ways. The formula for permutation with repetition can be used. n!/(p!r!s!)
a. 3+3+3+1= 4 ways b. 3+3+2+2=6 ways c. 3+3+2+1+1=30 ways d. 3+3+1+1+1+1=15 ways
e. 3+2+2+2+1=20 ways f. 3+2+2+1+1+1=60 ways g. 3+2+1+1+1+1+1=42 ways
h. 3+1+1+1+1+1+1+1=8 ways i. 2+2+2+2+2= 1 way j. 2+2+2+2+1+1=15 ways
k. 2+2+2+1+1+1+1=35 ways l. 2+2+1+1+1+1+1+1=28 ways m. 2+1+1+1+1+1+1+1+1= 9 ways
n. 1+1+1+1+1+1+1+1+1+1= 1 way
4+6+30+15++20+60+42+8+1+15+35+28+9+1= 274 ways