The word TOADSTOOL consists of the letters: T, O, A, D, S, T, O, O, L.

We need to select 5 letters including at least 2 Os and at least 1 T.

Consider the following cases:

1. Case 1: 2 Os and 1 T (OOT__). We need 2 more letters from A, D, S, L.

   The number of ways to choose 2 more letters from 4 is given by: \(\binom{4}{2} = 6\).

2. Case 2: 2 Os and 2 Ts (OOTT_). We need 1 more letter from A, D, S, L.

   The number of ways to choose 1 more letter from 4 is given by: \(\binom{4}{1} = 4\).

3. Case 3: 3 Os and 1 T (OOOT_). We need 1 more letter from A, D, S, L.

   The number of ways to choose 1 more letter from 4 is given by: \(\binom{4}{1} = 4\).

4. Case 4: 3 Os and 2 Ts (OOOTT). No more letters needed.

   There is only 1 way to choose this combination.

Adding all the cases together gives: \(6 + 4 + 4 + 1 = 15\).

The word STEEPLECHASE consists of the letters: S, T, E, E, P, L, E, C, H, A, S, E.

We need to select four letters including exactly one S. There are two S's in the word, so we choose one S and three other letters from the remaining 10 letters.

First, count the distinct letters after choosing one S: T, E, P, L, C, H, A, E, E, E.

We can choose the remaining three letters in the following ways:

1. Choose 3 E's: inom{6}{1} = 6
2. Choose 2 E's and 1 other letter: inom{6}{2} = 15
3. Choose 1 E and 2 other letters: inom{6}{3} = 20

Adding these scenarios gives the total number of selections:

\(6 + 15 + 20 = 41\)

Therefore, the total number of different selections is 42.

(iv) To find the number of selections with exactly one M and one A, we choose 1 M, 1 A, and 1 other letter from the remaining letters.

The word CAMERAMAN has the letters C, A, M, E, R, A, M, A, N.

There are 3 A's and 2 M's. We choose 1 M and 1 A, and then choose 1 more letter from the remaining 6 letters (C, E, R, N, A, M).

The number of ways to choose 1 letter from these 6 is given by:

\(\binom{4}{1} = 4\)

Thus, the number of different selections is 4.

(v) To find the number of selections with at least one M, we consider different cases:

1. Case 1: 1 M and 2 other letters (not both A's). Choose 1 M and 2 from the remaining 7 letters (C, A, E, R, A, N).

\(\binom{4}{2} = 6\)

2. Case 2: 2 M's and 1 other letter. Choose 1 from the remaining 7 letters (C, A, E, R, A, N).

\(\binom{4}{1} = 4\)

3. Case 3: 1 M, 1 A, and 1 other letter. Already calculated as 4.

\(Total number of selections = 6 + 4 + 4 = 16.\)

The word MISSISSIPPI consists of the following letters: M, I, S, S, I, S, S, I, P, P, I.

\(Count of each letter: M = 1, I = 4, S = 4, P = 2.\)

Total number of ways to choose 2 letters from 11: \(\binom{11}{2} = 55\).

Ways to choose 2 identical letters:

• 2 S's: \(\binom{4}{2} = 6\)
• 2 I's: \(\binom{4}{2} = 6\)
• 2 P's: \(\binom{2}{2} = 1\)

Total ways to choose 2 identical letters: \(6 + 6 + 1 = 13\).

Probability that the two letters are the same: \(\frac{13}{55}\).

(iii) To find the number of selections with exactly 2 Es and 2 Ns, we consider the letters in SEVENTEEN: S, E, V, E, N, T, E, E, N. We need to select 2 Es and 2 Ns, leaving 1 more letter to choose from S, V, or T. This can be done in 3 ways: EENN + S, EENN + V, EENN + T. Thus, there are 3 possible selections.

(iv) To find the number of selections with at least 2 Es, we consider the following cases:

1. 2 Es: Choose 2 Es and 3 other letters from S, V, N, T, N. This can be done in 7 ways: EENN + S, EENN + V, EENN + T, EENT + S, EENT + V, EENT + N, EENS + V.
2. 3 Es: Choose 3 Es and 2 other letters from S, V, N, T, N. This can be done in 7 ways: EEE + SV, EEE + SN, EEE + ST, EEE + VN, EEE + VT, EEE + NT, EEE + NN.
3. 4 Es: Choose 4 Es and 1 other letter from S, V, N, T. This can be done in 4 ways: EEEE + S, EEEE + V, EEEE + N, EEEE + T.

\(Adding these, the total number of ways is 7 + 7 + 4 = 18.\)

(iii) To find the number of selections with exactly 1 M and exactly 1 E, we first choose 1 M from the 2 Ms and 1 E from the 2 Es. This leaves us with 5 other letters (I, N, C, A, T) to choose 3 more letters from. The number of ways to choose 3 letters from these 5 is given by:

\(\binom{5}{3} = 10\)

(iv) To find the number of selections with at least 1 M and at least 1 E, we consider two methods:

Method 1: Consider separate groups:

• MME: Choose 2 Ms and 1 E, then choose 2 more letters from the remaining 5 letters (I, N, C, A, T): \(\binom{5}{2} = 10\)
• MEE: Choose 1 M and 2 Es, then choose 2 more letters from the remaining 5 letters: \(\binom{5}{2} = 10\)
• MMEE: Choose 2 Ms and 2 Es, then choose 1 more letter from the remaining 5 letters: \(\binom{5}{1} = 5\)

Summing these scenarios gives: \(10 + 10 + 5 = 25\)

Adding the scenario from part (iii) ME***: \(10\) (as calculated in part iii)

\(Total = 35\)

Method 2: Consider the criteria are met if ME are chosen:

Choose ME, then choose 3 more letters from the remaining 7 letters (I, N, C, A, T, M, E):

\(\binom{7}{3} = 35\)

Thus, the total number of selections is 35.

(i) To have exactly one 'L', we choose 1 'L' from the 2 available 'L's and then choose 2 more letters from the remaining 6 letters (C, O, I, D, E, R). This is calculated as:

\(\binom{2}{1} \times \binom{6}{2} = 2 \times 15 = 15\)

(ii) Without restrictions, we consider three cases: no 'L', one 'L', and two 'L's.

• No 'L': Choose 3 letters from the 6 letters (C, O, I, D, E, R):

\(\binom{6}{3} = 20\)

• One 'L': Already calculated in part (i) as 15.
• Two 'L's: Choose 1 more letter from the 6 letters (C, O, I, D, E, R):

\(\binom{6}{1} = 6\)

\(Total selections = 20 + 15 + 6 = 41\)

We need to select four letters from the word COPENHAGEN such that there is an equal number of Es and Ns, with at least one of each.

The word COPENHAGEN contains the letters: C, O, P, E, N, H, A, G, E, N.

There are 2 Es and 2 Ns in the word.

First, consider the case where we select one E and one N. We need to choose 2 more letters from the remaining 6 letters (C, O, P, H, A, G).

The number of ways to choose 2 letters from these 6 is given by the combination formula:

\(\binom{6}{2} = 15\)

So, there are 15 ways to select EN**.

Next, consider the case where we select both Es and both Ns (EENN). There is only 1 way to do this.

Therefore, the total number of different selections is:

\(15 + 1 = 16\)

(iv) The letters available without E are V, R, G, N. We need to select 3 letters with exactly 1 R. Possible selections are RVG, RVN, RGN. Thus, there are 3 ways.

(v) Without E, the letters are V, R, G, N. We need to select 3 letters:

• No Rs: Only one way, VGN.
• 2 Rs: Choose 1 more letter from V, G, N. Possible selections are RRV, RRG, RRN, which is 3 ways.

\(Total selections without E: 1 (no Rs) + 3 (2 Rs) = 4 ways.\)

The word VENEZUELA consists of the letters V, E, N, E, Z, U, E, L, A. There are three E's in the word.

To select four letters containing exactly one E, we first choose one E. This leaves us with six other letters: V, N, Z, U, L, A.

We need to choose three more letters from these six. The number of ways to choose 3 letters from 6 is given by the combination formula:

\(\binom{6}{3} = \frac{6!}{3!(6-3)!} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20\)

Thus, the number of possible selections containing exactly one E is 20.

(a) To form a 3-digit number with no repeated digits, we choose 3 different digits from the available numbers: 1, 2, 3, 4, 6. There are 5 different digits, so we calculate the permutations of choosing 3 out of 5:

\(^5P_3 = \frac{5!}{(5-3)!} = 5 \times 4 \times 3 = 60\)

Thus, there are 60 different numbers that can be made with no repeated digits.

(b) To form a number between 200 and 300, the first digit must be 2. The remaining two digits can be chosen from 1, 2, 3, 3, 4, 6, 6, 6. We list the possible numbers:

212, 213, 214, 216, 221, 223, 224, 226, 231, 232, 233, 234, 236, 241, 242, 243, 246, 261, 262, 263, 264, 266

There are 22 such numbers.

(iii) To select exactly 1 N and 1 A, we choose 1 N and 1 A from the letters TANZANIA. The remaining letters are T, Z, I, and the other A. We need to choose 2 more letters from these 3 letters. This can be done in \(\binom{3}{2} = 3\) ways.

(iv) To select exactly 1 N, we consider different cases based on the number of A's:

1. 0 A's: Select N and 3 other letters from T, Z, I, A. This is 1 way: N, T, Z, I.
2. 2 A's: Select N, 2 A's, and 1 other letter from T, Z, I. This can be done in \(\binom{3}{1} = 3\) ways.
3. 3 A's: Select N and 3 A's. This is 1 way: N, A, A, A.

Adding these, the total number of ways is \(1 + 3 + 1 = 8\) ways.

To solve this problem, we need to consider two cases: selecting 4 letters with exactly 2 Gs and selecting 4 letters with exactly 3 Gs.

Case 1: Exactly 2 Gs

We choose 2 Gs from the 3 available Gs. The remaining 2 letters must be chosen from the letters A, E, R, T. The possible combinations are:

• GG with AA
• GG with AE
• GG with EE
• GG with RA
• GG with RE
• GG with RT
• GG with TA
• GG with TE

This gives us 8 ways.

Case 2: Exactly 3 Gs

We choose 3 Gs from the 3 available Gs. The remaining letter must be chosen from A, E, R, T. The possible combinations are:

• GGG with A
• GGG with E
• GGG with R
• GGG with T

This gives us 4 ways.

Total

\(Adding both cases together, we have a total of 8 + 4 = 12 ways.\)

The word 'REMEMBRANCE' consists of the letters: R, E, M, E, M, B, R, A, N, C, E.

Removing M and R, we have the letters: E, E, B, A, N, C, E.

We need to choose 4 letters with at least 2 Es.

Case 1: Exactly 2 Es. Choose 2 more letters from B, A, N, C.

The number of ways to choose 2 Es from 3 Es is \(\binom{3}{2} = 3\).

The number of ways to choose 2 more letters from B, A, N, C is \(\binom{4}{2} = 6\).

Total for this case: \(3 \times 6 = 18\) ways.

Case 2: Exactly 3 Es. Choose 1 more letter from B, A, N, C.

The number of ways to choose 3 Es from 3 Es is \(\binom{3}{3} = 1\).

The number of ways to choose 1 more letter from B, A, N, C is \(\binom{4}{1} = 4\).

Total for this case: \(1 \times 4 = 4\) ways.

Adding both cases: \(18 + 4 = 22\) ways.

Alternatively, choose 2 letters from E, E, E, B, A, N, C (5 letters): \(\binom{5}{2} = 10\) ways.

Thus, the total number of ways is 10.

(ii) The word TELEPHONE has 9 letters, with the letters T, L, P, H, N, and O available when there are no Es. This gives us 6 letters to choose from. The number of ways to choose 4 letters from these 6 is given by the combination formula:

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

(iii) If there is exactly 1 E, we choose 1 E and then select 3 more letters from the remaining 6 letters (T, L, P, H, N, O). The number of ways to choose 3 letters from these 6 is:

\(\binom{6}{3} = 20\)

(iv) With no restrictions, we consider all possible selections of 4 letters from the 9 letters, accounting for the presence of Es. We calculate the number of ways for each scenario:

• No Es: \(\binom{6}{4} = 15\)
• 1 E: \(\binom{6}{3} = 20\)
• 2 Es: Choose 2 Es and 2 more letters from the remaining 6: \(\binom{6}{2} = 15\)
• 3 Es: Choose 3 Es and 1 more letter from the remaining 6: \(\binom{6}{1} = 6\)

Total number of ways: \(15 + 20 + 15 + 6 = 56\)