Suppose you are playing a word game with seven distinct letters. How many seven letter words can there be?
Hint:Think about scrabble.
It is ok it the word is gibberish in this case
7! = 7*6*5*4*3*2*1 = 5040
You get a different (smaller) answer if some of the letters are the same.
How did you get that?
The first letter can be 7 choices
the second letter can be 6 choices
the third letter can be 5 choices, and so on.
To calculate the number of seven-letter words that can be formed using seven distinct letters, we can use the concept of permutations.
A permutation is an arrangement of objects in a specific order. In this case, since we have seven distinct letters, we want to calculate the number of permutations of these letters to form a seven-letter word.
To do this, we can use the formula for permutations, which is:
P(n, r) = n! / (n - r)!
Where:
P(n, r) represents the number of permutations of n objects taken r at a time,
n! represents the factorial of n (the product of all positive integers less than or equal to n),
n - r represents the number of objects left after taking r at a time, and
(n - r)! represents the factorial of n - r.
So, in this case, we want to find P(7, 7) since we want to arrange all seven distinct letters to form a seven-letter word.
Using the formula:
P(7, 7) = 7! / (7 - 7)!
= 7! / 0!
= 7! / 1
= 7!
Since the factorial of 7 (7!) is the product of all positive integers from 1 to 7:
7! = 1 * 2 * 3 * 4 * 5 * 6 * 7
= 5040
So, there can be 5040 different seven-letter words that can be formed using seven distinct letters in this word game.