Given: A fair dice is tossed repeatedly until six shows up 3 times.
To find: The probability that exactly 5 tosses are needed.

Approach:
To get exactly 3 sixes in `n` rolls, it is necessary that the last roll is a six, and there are `2` sixes in the first `n-1` rolls. Therefore, the probability of getting exactly 3 sixes in `n` rolls is given by:
P(n) = (5/6)^(n-1) * (1/6) * C(n-1, 2)
where C(n-1, 2) denotes the number of ways of choosing 2 positions from the first `n-1` rolls for the `2` sixes.

Now, the probability of getting exactly 3 sixes in `n` rolls, given that the third six occurs on the `n`th roll, is given by:
P(n) = (5/6)^(n-4) * (1/6)^3 * C(n-1, 2)

The probability of getting exactly 3 sixes in the first `n` rolls is the sum of the probabilities of getting exactly 3 sixes, given that the third six occurs on the `n`th roll, for all `n >= 3`:
P = ∑ P(n), where the sum is taken over `n` from `3` to `∞`.

The probability that exactly 5 tosses are needed is the probability of getting exactly 3 sixes in the first 5 rolls, and no more than 3 sixes in the first 4 rolls:
P = P(5) - P(4)
P = (5/6)^4 * (1/6) * C(4,2) - (5/6)^3 * (1/6) * C(3,2)
P = 25/1296

Therefore, the correct option is (a) 25/1296.

Solution:

Let's break down the problem and solve it step by step.

Step 1: Probability of getting a six on a single dice toss

The probability of getting a six on a single dice toss is 1/6.

Step 2: Probability of getting three sixes in five tosses

To get three sixes in five tosses, we need to get sixes on exactly three tosses and non-sixes on the remaining two tosses. The probability of getting a six on a single toss is 1/6 and the probability of not getting a six is 5/6. Therefore, the probability of getting three sixes in five tosses is:

P(getting three sixes in five tosses) = (1/6)^3 * (5/6)^2 = 5/7776

Step 3: Probability of getting three sixes in six tosses

To get three sixes in six tosses, we can either get three sixes in the first five tosses and a six on the sixth toss or get two sixes in the first five tosses, a non-six on the sixth toss, and a six on the seventh toss. The probability of getting three sixes in the first five tosses and a six on the sixth toss is:

P(getting three sixes in the first five tosses and a six on the sixth toss) = (1/6)^3 * (5/6)^2 * 1/6 = 5/46656

The probability of getting two sixes in the first five tosses, a non-six on the sixth toss, and a six on the seventh toss is:

P(getting two sixes in the first five tosses, a non-six on the sixth toss, and a six on the seventh toss) = (1/6)^2 * (5/6)^3 * 1/6 * 1/6 = 25/279936

Therefore, the total probability of getting three sixes in six tosses is:

P(getting three sixes in six tosses) = P(getting three sixes in the first five tosses and a six on the sixth toss) + P(getting two sixes in the first five tosses, a non-six on the sixth toss, and a six on the seventh toss) = 5/46656 + 25/279936 = 5/7776

Step 4: Probability of getting three sixes in seven tosses

To get three sixes in seven tosses, we can either get three sixes in the first six tosses and a six on the seventh toss or get two sixes in the first six tosses, a non-six on the seventh toss, and a six on the eighth toss. The probability of getting three sixes in the first six tosses and a six on the seventh toss is:

P(getting three sixes in the first six tosses and a six on the seventh toss) = (1/6)^3 * (5/6)^3 * 1/6 = 25/279936

The probability of getting two sixes in the first six tosses, a non-six on the seventh toss, and a six on the eighth toss is:

P(getting two sixes in the first six tosses, a non-six on the seventh toss, and a six on the eighth toss)