Question

how many possible permutations are there in the letters of the word PHILIPPINES?

Answer 1

Let's make anagrams of the word:

p = 3 letters

h = 1 letter

i = 3 letters

l = 1 letter

n = 1 letter

e = 1 letter

s = 1 letter

The word Philippines has 11 letters, so let's do a permutation of 11.But we also have letters equal to "p" and "i", so we will make the following permutations for these letters 3! and 3! Let's see how it will look:

P_n^{\left(n_1,n_2)\right} = \dfrac{n!}{n_1!\:n_2!}

P_{11}^{\left(3,3)\right} = \dfrac{11!}{3!\:3!}

P_{11}^{\left(3,3)\right} = \dfrac{11*10*\diagup\!\!\!\!9^3*\diagup\!\!\!8^4*7*6*5*4*\diagup\!\!\!\!3!}{\diagup\!\!\!\!3^1*\diagup\!\!\!\!2^1*1*\diagup\!\!\!\!3!}

P_{11}^{\left(3,3)\right} = 11*10*3*4*7*6*5*4

\boxed{\boxed{P_{11}^{\left(3,3)\right} = 1108800\:anagrams\:or\:permutations\:of\:letters}}\end{array}}\qquad\checkmark

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Answer 2

Total number of permutations are there in the letters of the word  PHILIPPINES=1108800

Step-by-step explanation:

In the word PHILIPPINES

P repeated=3

I repeated=3

Total letters in the world=11

Permutation formula: When r repeated p times , s repeated q times in a word of total letters n is given by

$\frac{n!}{p!q!}$

Using the formula

Total number of permutations are there in the letters of the word  PHILIPPINES=$\frac{11!|}{3!3}=\frac{11\times 10\times 9\times 8\times 7\times 6\times 5\times 4\times 3!}{3\times 2\times 1\times 3!}=1108800$

Total number of permutations are there in the letters of the word  PHILIPPINES=1108800

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