Text Twist
In the game Text Twist, six letters are given and the player must form words of different lengths using those letters. Suppose the letters in a particular game are ENHSIC.


a. How many different arrangements are possible using all 6 letters?

Verified step by step guidance

1. Identify the total number of letters to be arranged. Here, there are 6 letters: E, N, H, S, I, C.
2. Since all letters are distinct, the number of different arrangements (permutations) of all 6 letters is given by the factorial of 6.
3. Recall that the factorial of a number \(n\), denoted by \(n!\), is the product of all positive integers from 1 to \(n\). So, \$6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1$.
4. Write the formula for the number of permutations of 6 distinct letters as \(P = 6!\).
5. To find the total number of different arrangements, calculate the value of \$6!$ (though the problem asks only for the method, not the final number).

Key Concepts

Permutations

Permutations refer to the number of ways to arrange a set of distinct objects in order. When all objects are used, the total permutations are calculated as the factorial of the number of objects (n!). For example, arranging 6 unique letters results in 6! possible arrangements.

Factorial Notation

Factorial, denoted by n!, is the product of all positive integers from 1 up to n. It is used to calculate permutations and combinations. For instance, 6! = 6 × 5 × 4 × 3 × 2 × 1 = 720, representing the total ways to arrange 6 distinct items.

Distinct Letters Assumption

When calculating permutations, it is important to confirm whether letters are distinct or repeated. If all letters are unique, the total permutations are simply n!. If letters repeat, the formula adjusts by dividing by factorials of repeated counts to avoid overcounting.