find the number of distinguishable arrangements of the letters of the word.\nmillion\nthere are…

find the number of distinguishable arrangements of the letters of the word.\nmillion\nthere are \\(\\square\\) distinguishable arrangements.\n(simplify your answer.)

find the number of distinguishable arrangements of the letters of the word.\nmillion\nthere are \\(\\square\\) distinguishable arrangements.\n(simplify your answer.)

Answer

Explanation:

Step1: Count total and repeated letters

The word "MILLION" has 7 letters. The letter 'L' appears 2 times, and other letters (M, I, O, N) appear once each.

Step2: Apply permutation formula for repeated elements

The formula for distinguishable permutations of a word with ( n ) total letters and ( n_1, n_2, \dots, n_k ) repeated letters is ( \frac{n!}{n_1! \cdot n_2! \cdot \dots \cdot n_k!} ). Here, ( n = 7 ), ( n_1 = 2 ) (for 'L'), and others are 1. So we calculate ( \frac{7!}{2!} ).

Step3: Calculate factorials

( 7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040 ), ( 2! = 2 \times 1 = 2 ).

Step4: Divide to get result

( \frac{5040}{2} = 2520 ).

Answer:

2520