Let $S$ be the set of all ternary strings defined over the alphabet $\{a, b, c\}$. Consider all strings in $S$ that contain at least one occurrence of two consecutive symbols, that is, “aa”, “bb” or “cc”. The number of such strings of length 5 that are possible is ________. (Answer in integer)

To find the number of ternary strings of length $5$ containing at least one occurrence of two consecutive identical symbols ("aa", "bb", or "cc"), we can use the complement principle:

1. Total number of ternary strings of length 5: Since the alphabet has $3$ symbols $(\{a, b, c\})$ and each position can take any of the $3$ symbols independently:

   $$\text{Total strings} = 3^5 = 243$$

2. Number of strings with NO two consecutive identical symbols:

   • For the $1^{\text{st}}$ position, there are $3$ choices.

   • For the $2^{\text{nd}}$ position, it cannot be equal to the $1^{\text{st}}$ symbol, leaving $2$ choices.

   • For the $3^{\text{rd}}$ position, it cannot be equal to the $2^{\text{nd}}$ symbol, leaving $2$ choices.

   • For the $4^{\text{th}}$ position, it cannot be equal to the $3^{\text{rd}}$ symbol, leaving $2$ choices.

   • For the $5^{\text{th}}$ position, it cannot be equal to the $4^{\text{th}}$ symbol, leaving $2$ choices.

   $$\text{Number of strings without consecutive identical symbols} = 3 \times 2^4 = 3 \times 16 = 48$$

3. Number of required strings:

   $$\text{Required strings} = 243 - 48 = 195$$

Thus, the correct answer is 195.