How many permutations of the word "TEACHER" are there if the vowels must appear in alphabetical order?

Here are the solutions for questions 13 and 14. Question 13: How many permutations of the word "BANANA" are there if the two N's cannot be adjacent? Step 1: Identify the letters and their frequencies in "BANANA". The word "BANANA" has 6 letters: B: 1 time A: 3 times N: 2 times Step 2: Calculate the total number of permutations of "BANANA" without any restrictions. The formula for permutations with repeated letters is (n!)/(p_1! p_2! ... p_k!). Here, n=6, A is repeated 3 times (p_1=3), and N is repeated 2 times (p_2=2). Total permutations = (6!)/(3!2!) = (720)/((6)(2)) = (720)/(12) = 60 Step 3: Calculate the number of permutations where the two N's are adjacent. Treat "NN" as a single block. The items to arrange are now: B, A, A, A, (NN). The total number of items to arrange is 5. The letter A is repeated 3 times. Permutations with N's adjacent = (5!)/(3!) = (120)/(6) = 20 Step 4: Subtract the "N's adjacent" case from the "total permutations" case to find when N's are not adjacent. Permutations (N's not adjacent) = Total permutations - Permutations (N's adjacent) Permutations (N's not adjacent) = 60 - 20 = 40 The number of permutations of the word "BANANA" where the two N's cannot be adjacent is 40. Question 14: How many permutations of the word "TEACHER" are there if the vowels must appear in alphabetical order? Step 1: Identify the letters, vowels, and consonants in "TEACHER". The word "TEACHER" has 7 letters. Vowels: A, E, E (3 vowels) Consonants: T, C, H, R (4 consonants) Step 2: Determine the required order for the vowels. The vowels A, E, E must appear in alphabetical order. This means their sequence must be A, E, E. Step 3: Choose positions for the vowels. There are 7 total positions for the letters. We need to choose 3 of these positions for the vowels. The number of ways to choose 3 positions out of 7 is given by the combination formula nk: 73 = (7!)/(3!(7-3)!) = (7!)/(3!4!) = (7 × 6 × 5)/(3 × 2 × 1) = 35 Step 4: Place the vowels in the chosen positions. Once the 3 positions are chosen, the vowels A, E, E must be placed in them in the specific alphabetical order (A, E, E). Since the two E's are identical, there is only 1 unique way to place them in the chosen slots (e.g., the first chosen slot gets A, the second gets E, the third gets E). Step 5: Arrange the consonants in the remaining positions. There are 4 remaining positions and 4 consonants (T, C, H, R). All these consonants are distinct. The number of ways to arrange 4 distinct consonants in 4 distinct positions is 4!. 4! = 4 × 3 × 2 × 1 = 24 Step 6: Calculate the total number of permutations. Multiply the number of ways to choose positions for vowels, the number of ways to place vowels, and the number of ways to arrange consonants. Total permutations = (Ways to choose vowel positions) × (Ways to place vowels) × (Ways to arrange consonants) Total permutations = 35 × 1 × 24 = 840 The number of permutations of the word "TEACHER" where the vowels must appear in alphabetical order is 840. That's 2 down. 3 left today — send the next one.