How many arrangements of all the letters in the word WINNIPEG can be created if the N's cannot be together?
To find the number of arrangements of the letters in the word WINNIPEG where the N's cannot be together, we can use the principle of counting and permutation.
First, let's consider the total number of arrangements if there were no restrictions. The word WINNIPEG has 8 letters, including 2 N's. So, there are 8 positions to fill with letters.
The number of total arrangements is given by the formula for permutations, which is calculated as:
Total arrangements = 8! / (2! x 2!) = 8! / 4
Next, let's find the number of arrangements where the N's are together. We can consider the two N's as a single entity. So, we have 7 entities to arrange: (WNIGPEN). The number of arrangements with the N's together is:
Arrangements with N's together = 7!
Now, let's subtract the number of arrangements with the N's together from the total number of arrangements to find the number of arrangements where the N's cannot be together:
Arrangements without N's together = Total arrangements - Arrangements with N's together
= 8! / 4 - 7!
By calculating this expression, we can find the number of arrangements of the letters in the word WINNIPEG where the N's cannot be together.
To find the number of arrangements of the letters in the word WINNIPEG where the N's cannot be together, we can use the principle of permutations.
First, we count the total number of possible arrangements without any restrictions. The word WINNIPEG has 8 letters, so there are 8! (8 factorial) arrangements in total.
Now, let's consider the N's as a single entity. We can treat them as one letter. So, we have 6 distinct entities: W, I, N, N (combined), I, P, E, and G. Within these entities, there are 3 distinct I's.
The total number of arrangements where the N's are together can be calculated as 7! * 3! since the N's can be arranged among themselves in 3! ways.
Therefore, the total number of arrangements in which the N's cannot be together is: 8! - (7! * 3!) = 40,320 - (5,040 * 6) = 40,320 - 30,240 = 10,080.
So, there are 10,080 different arrangements of the letters in the word WINNIPEG where the N's cannot be together.