in how many ways can the letters in the word balloon be arranged?\n210\n1,260\n2,520\n5,040
in how many ways can the letters in the word balloon be arranged?\n210\n1,260\n2,520\n5,040
Answer
Explanation:
Step1: Count the total number of letters
The word "balloon" has (7) letters.
Step2: Count the repeated letters
The letter 'l' appears (2) times and the letter 'o' appears (2) times.
Step3: Use the permutation formula for repeated elements
The formula for permutations of a multi - set is (\frac{n!}{n_1!n_2!\cdots n_k!}), where (n) is the total number of elements, and (n_i) are the number of times each repeated element appears. Here (n = 7), (n_1=2) (for 'l') and (n_2 = 2) (for 'o'). [ \begin{align*} \frac{7!}{2!2!}&=\frac{7\times6\times5\times4\times3\times2\times1}{(2\times1)\times(2\times1)}\ &=\frac{5040}{4}\ & = 1260 \end{align*} ]
Answer:
1,260