The number of 4 letter words

Question:

The number of 4 letter words (with or without meaning) that can be formed from the eleven letters of the word 'EXAMINATION' is_______.

Solution:

EXAMINATION

$2 \mathrm{~N}, 2 \mathrm{~A}, 2 \mathrm{I}, \mathrm{E}, \mathrm{X}, \mathrm{M}, \mathrm{T}, \mathrm{O}$

Case I : If all are different, then

${ }^{8} p_{4}=\frac{8 !}{4 !}=8.7 .6 .5=1680$

Case II : If two are same and two are different, then

${ }^{3} C_{1} \cdot{ }^{7} C_{2} \cdot \frac{4 !}{2 !}=3.21 .12=756$

Case III : If two are same and other two are same, then

${ }^{3} C_{2} \cdot \frac{4 !}{2 ! 2 !}=3.6=18$

$\therefore \quad$ Total cases $=1680+756+18=2454$

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