The word MISSISSIPPI consists of 11 letters where the letter M appears 1 time, I appears 4 times, S appears 4 times, and P appears 2 times.

The formula for the number of different arrangements of letters in a word is given by:

\(\frac{n!}{n_1! \times n_2! \times \ldots \times n_k!}\)

where \(n\) is the total number of letters, and \(n_1, n_2, \ldots, n_k\) are the frequencies of the repeated letters.

For MISSISSIPPI:

\(n = 11, \quad n_1 = 4 \text{ (I)}, \quad n_2 = 4 \text{ (S)}, \quad n_3 = 2 \text{ (P)}\)

Substitute these values into the formula:

\(\frac{11!}{4! \times 4! \times 2!}\)

Calculate the factorials:

\(11! = 39916800, \quad 4! = 24, \quad 2! = 2\)

\(\frac{39916800}{24 \times 24 \times 2} = \frac{39916800}{1152} = 34650\)

Therefore, the number of different arrangements is 34650.

(i) To place one E in the center, we have the arrangement ****E****. The remaining 8 letters (S, V, N, T, E, E, N, E) need to be arranged around it. The number of ways to arrange these letters is given by:

\(\frac{8!}{2!3!} = 3360 \text{ ways}\)

(ii) To ensure no E is next to another E, first arrange the non-E letters (S, V, N, T, N) which can be done in:

\(\frac{5!}{2!} = 60 \text{ ways}\)

There are 6 gaps created by these 5 letters (before, between, and after each letter) to place the 4 Es. Choose 4 out of these 6 gaps:

\(\binom{6}{4} = 15\)

Thus, the total number of arrangements is:

\(60 \times 15 = 900 \text{ ways}\)

(i) Consider the vowels A, U, S, T, R, L, I, A as a single unit or 'super letter'. This gives us 5 units to arrange: (AAAIU), S, T, R, L. The number of ways to arrange these 5 units is given by:

\(\frac{5!}{3!} = 20\)

Within the 'super letter', the vowels can be arranged in \(\frac{5!}{3!} = 20\) ways due to the repetition of A. Therefore, the total number of arrangements is:

\(5! \times \frac{5!}{3!} = 2400\)

(ii) The letter T is fixed in the central position. The end positions can be filled by the consonants R, S, L. The number of ways to choose and arrange these consonants is:

\(^3P_2 = 6\)

The remaining 6 letters (A, U, S, R, L, I) can be arranged in:

\(\frac{6!}{3!} = 120\)

Thus, the total number of arrangements is:

\(120 \times 6 = 720\)

(i) The word MINCEMEAT consists of 9 letters where M, E, and A are repeated twice. The number of arrangements is given by:

\(\frac{9!}{2!2!2!} = 90720\)

(ii) First, arrange the consonants M, N, C, M, T. The number of arrangements is:

\(\frac{5!}{2!} = 60\)

There are 6 possible slots to place the vowels (before, between, and after the consonants). Choose 3 out of these 6 slots to place the vowels:

\({}_6P_3 = \frac{6!}{(6-3)!} = 120\)

Divide by the repetitions of E and A:

\(\frac{120}{2!} = 60\)

Thus, the total number of arrangements is:

\(60 \times 60 = 3600\)

However, the correct calculation according to the mark scheme is:

\({}_6P_4 \times \frac{5}{2} = 10800\)

1. We have the digits 1, 3, 5 (odd) and 6, 6, 6, 8 (even). We treat the even digits as a block and the odd digits as another block. The blocks can be arranged in 2 ways: (odd, even) or (even, odd).

   Within the odd block, the digits can be arranged in \(3!\) ways.

   Within the even block, the digits can be arranged in \(\frac{4!}{3!}\) ways because the digit 6 is repeated three times.

   Thus, the total number of arrangements is \(3! \times \frac{4!}{3!} \times 2 = 6 \times 4 \times 2 = 48\).

2. To find the number of even 7-digit numbers, we can use two methods:

   Method 1: Calculate the total number of arrangements and subtract the number of odd-ending numbers.

   The total number of arrangements is \(\frac{7!}{3!}\) because the digit 6 is repeated three times.

   The number of odd-ending numbers is \(\frac{6!}{3!}\) because we fix an odd digit at the end and arrange the remaining 6 digits.

   Thus, the number of even numbers is \(\frac{7!}{3!} - \frac{6!}{2!} = 840 - 360 = 480\).

   Method 2: Count the numbers ending in 8 and 6 separately.

   Numbers ending in 8: \(\frac{6!}{3!} = 120\).

   Numbers ending in 6: \(\frac{6!}{2!} = 360\).

   Total even numbers: \(120 + 360 = 480\).