How many ways can 6 different letters be arranged if 2 letters are always togeth

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Question: How many ways can 6 different letters be arranged if 2 letters are always together?

Options:

Correct Answer: 1440

Solution:

Treat the 2 letters as one unit. So, we have 5 units to arrange: 5! = 120. The 2 letters can be arranged in 2! = 2 ways. Total = 120 * 2 = 240.

How many ways can 6 different letters be arranged if 2 letters are always together?

How many ways can 6 different letters be arranged if 2 letters are always together?

Step 1: Identify the 6 different letters. Let's call them A, B, C, D, E, and F.
Step 2: Since 2 letters must always be together, choose those 2 letters. For example, let's say we choose A and B.
Step 3: Treat the 2 letters (A and B) as one single unit or block. Now, instead of 6 letters, we have 5 units to arrange: (AB), C, D, E, F.
Step 4: Calculate the number of ways to arrange these 5 units. The formula for arranging n units is n!. So, we calculate 5! (which is 5 factorial).
Step 5: Calculate 5! = 5 × 4 × 3 × 2 × 1 = 120.
Step 6: Now, consider the arrangement of the 2 letters (A and B) within their block. They can be arranged in 2 ways: AB or BA.
Step 7: Multiply the number of arrangements of the 5 units (120) by the arrangements of the 2 letters (2). So, 120 × 2 = 240.
Step 8: The total number of ways to arrange the 6 letters with the condition that 2 letters are always together is 240.

Permutations with Restrictions – This concept involves arranging items with certain conditions, such as keeping specific items together.
Factorial Calculations – Understanding how to calculate permutations using factorials is essential for solving arrangement problems.