Question

How many different 10 letter words (real or imaginary) can be formed from the following letters

H,T,G,B,X,X,T,L,N,J

Answer 1

Consider the following letters:

$$ \mathrm{H}, \mathrm{T}, \mathrm{G}, \mathrm{B}, \mathrm{X}, \mathrm{X}, \mathrm{T}, \mathrm{L}, \mathrm{N}, \mathrm{J} $$

Find different 10 letter words can be formed from the above letters.

Number of letters in the word \(=10\)

The number of repititions of the letters in the respective word \(=2 ! 2 !\)

Recall the formula,

The total number of words in a letters \(=\frac{n !}{a ! b !}\)

Here \(n=\) Number of letters in the word.

\(a \mid b !=\) The number of repititions of the letters in the respective word. Given letters are \(\mathrm{H}, \mathrm{T}, \mathrm{G}, \mathrm{B}, \mathrm{X}, \mathrm{X}, \mathrm{T}, \mathrm{L}, \mathrm{N}, \mathrm{J}\)

Number of letters in the word 'n' \(=10\)

$$ \begin{array}{l} a=2 !(\text { Letter } T \text { repeating } 2 \text { times }) \\ b=2 !(\text { Letter } X \text { repeating } 2 \text { times }) \\ \text { Therefore, total number of words }=\frac{10 !}{2 ! 2 !} \\ =\frac{10 \cdot 9 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}{(2 \cdot 1)(2 \cdot 1)} \\ =907200 \end{array} $$

Thus, the total number of words is 907200