The word “balloon” consists of 7 letters where certain letters are repeated. Specifically, we have:
- b: 1 time
- a: 1 time
- l: 2 times
- o: 2 times
- n: 1 time
To find the number of distinct arrangements of the letters in the word “balloon,” we can use the formula for permutations of a multiset:
P(n; n1, n2, n3, … , nk) = n! / (n1! * n2! * n3! * … * nk!)
Where:
- n is the total number of items to arrange.
- n1, n2, …, nk are the frequencies of the repeated items.
In the case of “balloon”:
- n = 7 (total letters)
- n1 = 1 (for b)
- n2 = 1 (for a)
- n3 = 2 (for l)
- n4 = 2 (for o)
- n5 = 1 (for n)
Plugging in the values into the formula gives us:
Number of arrangements = 7! / (1! * 1! * 2! * 2! * 1!)
This calculation simplifies as follows:
- 7! = 5040
- 1! = 1
- 2! = 2
So, we proceed with:
Number of arrangements = 5040 / (1 * 1 * 2 * 2 * 1) = 5040 / 4 = 1260
Thus, the letters in the word “balloon” can be arranged in 1260 distinct ways.