Question

Сколько есть способов расставить в ряд буквы А, А, Б, Б, В и Е так, чтобы выполнялись все условия, приведённые ниже? Либо буквы А стоят рядом, либо буквы Б стоят рядом. На первом или последнем месте стоит гласная.

Answer 1

13
AAББВЕ
ААБВБЕ
ААВББЕ
ну и так далее

Answer 2

Calculating the Number of Ways to Arrange Letters

To calculate the number of ways to arrange the letters A, A, B, B, V, and E in a row such that either the A's are next to each other or the B's are next to each other and a vowel is at the first or last position, we can follow these steps:

1. Calculate the total number of arrangements. 2. Calculate the number of arrangements where the A's are next to each other. 3. Calculate the number of arrangements where the B's are next to each other. 4. Subtract the arrangements where both A's and B's are next to each other to avoid double counting. 5. Calculate the number of arrangements where a vowel is at the first or last position.

Let's calculate each of these steps.

Total Number of Arrangements

The total number of arrangements of the letters A, A, B, B, V, and E is given by the formula: \[ \frac{6!}{2! \times 2!} = 180 \]

Arrangements with A's Next to Each Other

The number of arrangements where the A's are next to each other can be calculated by treating the pair of A's as a single entity. This reduces the number of entities to arrange to 5. The number of arrangements where the A's are next to each other is: \[ 5! \times 2! = 240 \]

Arrangements with B's Next to Each Other

Similarly, the number of arrangements where the B's are next to each other is: \[ 5! \times 2! = 240 \]

Subtracting Double Counted Arrangements

We need to subtract the arrangements where both A's and B's are next to each other to avoid double counting. The number of arrangements where both A's and B's are next to each other is: \[ 4! \times 2! \times 2! = 48 \]

Arrangements with a Vowel at the First or Last Position

The number of arrangements where a vowel is at the first or last position can be calculated by treating the vowel at the first or last position as a single entity. This reduces the number of entities to arrange to 5. The number of arrangements where a vowel is at the first or last position is: \[ 5! \times 2 = 240 \]

Final Calculation

Now, we can calculate the total number of valid arrangements by summing the arrangements with A's next to each other, arrangements with B's next to each other, arrangements with a vowel at the first or last position, and then subtracting the double counted arrangements: \[ 240 + 240 + 240 - 48 = 672 \]

Therefore, there are 672 ways to arrange the letters A, A, B, B, V, and E in a row such that either the A's are next to each other or the B's are next to each other and a vowel is at the first or last position.