Find the number of words formed by permuting all the letters of the following words:
Question:

Find the number of words formed by permuting all the letters of the following words:

(i) INDEPENDENCE

(ii) INTERMEDIATE

(iii) ARRANGE

(iv) INDIA

(v) PAKISTAN

(vi) RUSSIA

(vii) SERIES

(viii) EXERCISES

(ix) CONSTANTINOPLE

Solution:

(i) This word consists of 12 letters that include three Ns, two Ds and four Es.

The total number of words is the number of arrangements of 12 things, of which 3 are similar to one kind, 2 are similar to the second kind and 4 are similar to the third kind.

$\Rightarrow \frac{12 !}{3 ! 2 ! 4 !}=1663200$

(ii) This word consists of 12 letters that include two Is, two Ts and three Es.

The total number of words is the number of arrangements of 12 things, of which 2 are similar to one kind, 2 are similar to the second kind and 3 are similar to the third kind.

$\Rightarrow \frac{12 !}{2 ! 2 ! 3 !}=19958400$

(iii) This word consists of 7 letters that include two Rs, and two As.

The total number of words is the number of arrangements of 7 things, of which 2 are similar to one kind and 2 are similar to the second kind.

$\Rightarrow \frac{7 !}{2 ! 2 !}=1260$

(iv) This word consists of 5 letters that include two Is.

The total number of words is the number of arrangements of 5 things, of which 2 are similar to one kind.

$\Rightarrow \frac{5 !}{2 !}=60$

(v) This word consists of 8 letters that include two As.

The total number of words is the number of arrangements of 7 things, of which 2 are similar to one kind.

$\Rightarrow \frac{8 !}{2 !}=20160$

(vi) This word consists of 6 letters that include two Ss.

The total number of words is the number of arrangements of 6 things, of which 2 are similar to one kind.

$\Rightarrow \frac{6 !}{2 !}=360$

(vii) This word consists of 6 letters that include two Ss and two Es.

The total number of words is the number of arrangements of 6 things, of which 2 are similar to one kind and 2 are similar to the second kind.

$\Rightarrow \frac{6 !}{2 ! 2 !}=180$

(viii) This word consists of 9 letters that include three Es and two Ss.

The total number of words is the number of arrangements of 9 things, of which 2 are similar to one kind and 2 are similar to the second kind.

$\Rightarrow \frac{9 !}{2 ! 3 !}=30240$

(ix) This word consists of 14 letters that include three Ns, two Os and two Ts.

The total number of words is the number of arrangements of 14 things, of which 3 are similar to one kind, 2 are similar to the second kind and 2 are similar to the third kind.

$\Rightarrow \frac{14 !}{3 ! 2 ! 2 !}=\frac{14 !}{24}$

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