The number of sequences of ten terms, whose terms are either 0 or 1 or 2 , that contain exactly five 1 s and exactly three 2 s , is equal to :
Solutions · 1
Official solution
To find the number of sequences of ten terms, each being either 0, 1, or 2, containing exactly five 1s and exactly three 2s, follow these steps:
Determine the Remaining Terms: Since there are 5 ones and 3 twos, you will need 2 zeros to fill the sequence (because 5+3+2=10).
Calculate the Number of Arrangements: You have a total of 10 positions to fill with these numbers (5 ones, 3 twos, 2 zeros). The formula for calculating permutations of a multiset is:
5!×3!×2!10!
Where:
10! is the factorial of the total number of terms.
5! is the factorial for the number of 1s.
3! is the factorial for the number of 2s.
2! is the factorial for the number of 0s.
Result: Simplifying the calculation gives:
5!×3!×2!10!=2520
Thus, there are 2520 possible sequences with the given conditions.