Calculating MINING Letter Arrangements

To find the number of distinct ways to arrange the letters in the word MINING, we need to account for the repeated letters.

Identify Letters and Repetitions

• The word MINING has 6 letters in total.
• The letters are M, I, N, I, N, G.
• The letter 'I' appears 2 times.
• The letter 'N' appears 2 times.
• The letters 'M' and 'G' appear 1 time each.

Apply Permutation Formula

The formula for permutations with repetitions is:

Where:

• $n$ is the total number of letters.
• $n_1, n_2, ..., n_k$ are the frequencies of each distinct repeated letter.

Perform the Calculation

In this case, $n = 6$. The frequencies of the repeated letters are 2 for 'I' and 2 for 'N'.

Number of arrangements = $ \frac{6!}{2! \times 2!} $

• Calculate $6!$: $ 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720 $
• Calculate $2!$: $ 2! = 2 \times 1 = 2 $
• Substitute the values into the formula: $ \frac{720}{2 \times 2} = \frac{720}{4} $
• Final calculation: $ \frac{720}{4} = 180 $

Therefore, there are 180 distinct ways to arrange the letters in the word MINING.