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:
$ \frac{n!}{n_1! n_2! ... n_k!} $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.
Important Questions from Permutations
How many words can be formed with the letters of the word 'POSTMAN', if every word begins with T and ends with M?
In how many ways can cells in a $3 \times 3$ grid be shaded, such that each row and each column have exactly one shaded cell? An example of one valid shading is shown.
- Three husband-wife pairs are to be seated at a circular table that has six identical chairs. Seating arrangements are defined only by the relative position of the people. How many seating arrangements are possible such that every husband sits next to his wife?
- The number of 'three-digit numbers' that can be formed using the digits from 1 to 9 without the repetition of each digit is ________.
- A box containing 10 identical compartments has 6 red balls and 2 blue balls. If each compartment can hold only one ball, then the number of different possible arrangements are
