Question

How many ternary strings of length 2 n are there in which the zeroes appear only in odd-numbered positions?

Answer 1

The number of ternary strings of length 2n with zeroes in odd-numbered positions is
`2^n * 3^n.`

Step-by-step explanation:

To find the number of ternary strings of length 2n in which the zeroes appear only in odd-numbered positions, we can first consider the even-numbered positions.

For any even-numbered position, we have 3 choices - 1, 2, or 3.

For odd-numbered positions, we can only choose from the set {1, 3}.

Since there are n even-numbered positions and n odd-numbered positions, we multiply the number of choices for each position together to get the total number of ternary strings.

Therefore, the number of ternary strings of length 2n with zeroes in odd-numbered positions is 2n * 3n.

Answer 2

In a string of length
`2n`, there are exactly
`n` odd-numbered posiitons. In the remaning
`n` even-numbered positions, we want to choose from only two possible digits, either a 1 or a 2. This means there are
`2^n` such ternary strings.