Permutations And Combinations

We need to select 6 persons including only 1 teacher from 4 teachers and 7 students.
Step 1: First, select 1 teacher from 4 teachers. This can be done in:
Step 2: Next, select 5 students from 7 students. This can be done in:
Step 3: The total number of selections is: Thus, the number of selections is 84.

The word "PERMUTATIONS" has 11 letters in total. The vowels in "PERMUTATIONS" are E, U, A, I, O, which are 5 vowels.
Step 1: Treat all the vowels as a single unit. This reduces the problem to arranging the 7 consonants and the unit of vowels. Thus, we have 7 + 1 = 8 units to arrange. The number of ways to arrange these 8 units is:
Step 2: Within the unit of vowels, the vowels can be arranged in ways:
Step 3: Since the letter "T" repeats twice in "PERMUTATIONS", we divide by to account for the repetition: Thus, the total number of arrangements is 2419200.

Let the number of children be .
Step 1: Each child gives a gift to every other child, so the total number of gifts is given by the combination formula:
Step 2: Set this equal to the total number of gifts:
Step 3: Multiply both sides by 2:
Step 4: Solve the quadratic equation: Using the quadratic formula: Approximating the square root of 1057, we get: Thus, the number of children is 12.