HAPPY HOLIDAYS Arrangements and Fraction Calculation

[Jameson]

Let A be the number of distinct arrangements of "HAPPY HOLIDAYS" (don't consider the space to be a character).

What is [math]A \times \frac{503}{97297200}[/math]?
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[Jameson]

Congratulations to the following members for their correct solution:

1) Sudharaka
2) MarkFL
3) soroban
4) veronica1999

Solution (from Sudharaka):[sp]The following is the list of letters in "Happy Holidays" and the number of times each letter appears.

H - 2, A - 2, P - 2, Y - 2, O - 1, L - 1, I - 1, D - 1, S - 1

Number of letters in "Happy Holidays" = 13

Therefore the number of possible arrangements (A) = \(\dfrac{13!}{(2!)^4}\)

\[\therefore A \times \frac{503}{97297200}=\frac{13!}{(2!)^4}\times\frac{503}{97297200}=2012\][/sp]