Final answer:
The number of 16-length strings that can be made from 10 Ls, 2 Ts, and 4 Ws, ensuring no two Ws are next to each other, is found using combinations to place Ws first, resulting in 715 different strings.
Step-by-step explanation:
The question relates to combinatorial mathematics and asks:
How many 16 length strings can be made from 10 Ls, 2 Ts and 4 Ws such that no 2 Ws are next to each other?
First, let's place the Ws, considering the restriction that no two Ws can be adjacent. Imagine there are 12 different slots created by the Ls and Ts (since 10 Ls + 2 Ts = 12 letters and each letter creates a new slot to its right). These slots include the beginning and end of the string, as well as between each letter. We can place the 4 Ws in these 13 potential positions (12 slots + 1 for the end).
To calculate the number of ways to place the 4 Ws in these 13 slots without any two being adjacent, we can use combinations:
- There will be C(13, 4) ways to choose slots for Ws.
After placing Ws, we will have 12 remaining slots to place the Ls and Ts. Because all Ls and Ts are indistinguishable among themselves:
- The remaining 10 Ls can be arranged in 1 way, since their order doesn't matter.
- Similarly, the 2 Ts can be arranged in 1 way.
Therefore, the number of different 16-length strings that can be formed, with the given conditions, is the product of these possibilities: C(13, 4). The calculation of C(13, 4), which represents the combination of 13 slots taken 4 at a time, will give us the answer.
The value of C(13, 4) is calculated as:
C(13, 4) = 13! / (4!(13 - 4)!) = (13 × 12 × 11 × 10) / (4 × 3 × 2 × 1) = 715
Thus, there are 715 different strings that can be made with the given conditions.