Number of Combinations of the Letters of the Word COLLEGE

To find the number of combinations of the letters of the word "COLLEGE," we can use the concept of permutations and combinations.

Permutations and Combinations:

Permutations and combinations are mathematical techniques used to count the number of ways objects can be arranged or selected.

Permutations: Permutations refer to the arrangements of objects in a specific order. The formula to calculate the number of permutations is given by:

P(n, r) = n! / (n - r)!

Where P(n, r) represents the number of permutations of selecting r objects from a total of n objects, and "!" denotes the factorial of a number.

Combinations: Combinations refer to the selections of objects without considering the order. The formula to calculate the number of combinations is given by:

C(n, r) = n! / (r! * (n - r)!)

Where C(n, r) represents the number of combinations of selecting r objects from a total of n objects.

Analysis of the Word "COLLEGE":

The given word "COLLEGE" consists of 7 letters. To find the number of combinations, we need to consider all possible arrangements and selections of these letters.

Arrangements:

To find the number of arrangements, we can use the formula for permutations. In this case, we want to arrange all 7 letters of the word "COLLEGE."

P(7, 7) = 7! / (7 - 7)!
= 7! / 0!
= 7! / 1
= 7!

Since any number factorial divided by 1 is the number itself, we have:

7! = 7 * 6 * 5 * 4 * 3 * 2 * 1
= 5040

Therefore, there are 5040 different arrangements of the letters in the word "COLLEGE."

Selections:

To find the number of selections, we can use the formula for combinations. In this case, we want to select various numbers of letters from the word "COLLEGE."

C(7, 1) = 7! / (1! * (7 - 1)!)
= 7! / (1! * 6!)
= 7! / 6!
= 7

C(7, 2) = 7! / (2! * (7 - 2)!)
= 7! / (2! * 5!)
= 7 * 6 / 2
= 21

C(7, 3) = 7! / (3! * (7 - 3)!)
= 7! / (3! * 4!)
= 35

Continuing this process, we can find the number of selections for various values of r.

Therefore, the number of combinations of the letters of the word "COLLEGE" can be found by summing up the number of selections for each value of r:

C(7, 1) + C(7, 2) + C(7, 3) + C(7, 4) + C(7, 5) + C(7