A bag contains 5 cards with the letter 'A', 3 cards with the letter 'B' and 2 cards with the letter 'C'. $\text{A } \text{A } \text{A } \text{A } \text{A } \text{B } \text{B } \text{B } \text{C } \text{C}$ The 10 cards are arranged at random in a row to form a letter sequence. For example, AABBCAAACB is a possible letter sequence.

1. Apply the formula for permutations of non-distinct objects

The sequence consists of 10 letters: 5 'A's, 3 'B's, and 2 'C's.
$\text{Number of possible sequences} = \frac{10!}{5! \times 3! \times 2!}$

2. Evaluate the factorial expression

$\frac{10 \times 9 \times 8 \times 7 \times 6}{3 \times 2 \times 1 \times 2 \times 1} = \frac{30240}{12} = 2520$

1. Find the total number of letter sequences where no two 'B's are next to each other.

First, we arrange the 5 'A's and 2 'C's.
Number of ways $= \frac{7!}{5!\,2!} = 21$

These 7 letters create 8 spaces (including the ends) where the 'B's can be placed.
To ensure no two 'B's are adjacent, we choose 3 of these 8 spaces for the 3 'B's.
Number of ways $= \binom{8}{3} = 56$

Number of sequences with separated 'B's $= 21 \times 56 = 1176$

2. Find the number of sequences where no two 'B's are next to each other, but the two 'C's ARE next to each other.

We group the two 'C's together as a single block CC.
We first arrange the 5 'A's and the 1 CC block.
Number of ways $= \frac{6!}{5!\,1!} = 6$

These 6 items create 7 spaces (including the ends).
We choose 3 of these 7 spaces to place the 3 'B's (to ensure 'B's remain separated).
Number of ways $= \binom{7}{3} = 35$

Number of sequences with separated 'B's and grouped 'C's $= 6 \times 35 = 210$

3. Subtract to find the number of sequences satisfying both conditions.

Number of sequences where no two 'B's are adjacent AND no two 'C's are adjacent
$= 1176 - 210 = 966$

1. Find the probability of the given condition (that the second letter is not an 'A').

Let $X$ be the event that the second letter is not 'A'.
There are 5 'A's, 3 'B's, and 2 'C's (total 10 cards).

$P(X) = P(\text{2nd is 'B'}) + P(\text{2nd is 'C'})$
$P(X) = \frac{3}{10} + \frac{2}{10} = \frac{1}{2}$

2. Find the probability that the first two letters are identical AND the second letter is not an 'A'.

Let $Y$ be the event that the first two letters are identical.
The event $Y \cap X$ occurs if the first two letters are both 'B' or both 'C'.

$P(Y \cap X) = P(\text{BB}) + P(\text{CC})$
$P(Y \cap X) = \left(\frac{3}{10} \times \frac{2}{9}\right) + \left(\frac{2}{10} \times \frac{1}{9}\right)$
$P(Y \cap X) = \frac{6}{90} + \frac{2}{90} = \frac{8}{90} = \frac{4}{45}$

3. Apply the conditional probability formula to find the required probability.

$P(Y \mid X) = \frac{P(Y \cap X)}{P(X)}$
$P(Y \mid X) = \frac{\frac{4}{45}}{\frac{1}{2}}$
$P(Y \mid X) = \frac{4}{45} \times \frac{2}{1} = \frac{8}{45}$

1. Find the total number of ways to arrange the 10 cards in a circle, treating all identical cards as distinct to ensure equiprobable outcomes.

$\text{Total arrangements} = (10 - 1)! = 9!$

2. Find the number of favourable arrangements where no two 'A's are adjacent.

First, arrange the 5 non-'A' cards (3 'B's and 2 'C's) in a circle:
$\text{Ways to arrange non-'A' cards} = (5 - 1)! = 4!$
This creates exactly 5 spaces between them. To ensure no two 'A's are next to each other, the 5 'A' cards must be placed into these 5 spaces:
$\text{Ways to arrange 'A' cards} = 5!$
$\text{Total favourable arrangements} = 4! \times 5!$

3. Calculate the required probability.

$\text{Probability} = \frac{4! \times 5!}{9!} = \frac{24 \times 120}{362880} = \frac{1}{126}$