Q. The number of five-letter words formed with the letters of the word CALCULUS is_____.
Permutations and Combinations
Solution:
$2 \,\,\,\, 1 \,\,\,\,\,2 \,\,\,\,2 \,\,\, 1 $
$C \,\,A \,\, L \,\, U \,\,S $
Type of word
Number of permutations
2 identical +2 identical +1 different
${ }^{3} C _{2} \cdot{ }^{3} C _{1} \cdot \frac{5 !}{2 ! 2 ! 1 !}=270$
2 identical +3 different
${ }^{3} C _{1} \cdot{ }^{4} C _{3} \cdot \frac{5 !}{2 !}=720$
All 5 different
120
$\therefore $ Total number of word $=1110$
| Type of word | Number of permutations |
|---|---|
| 2 identical +2 identical +1 different | ${ }^{3} C _{2} \cdot{ }^{3} C _{1} \cdot \frac{5 !}{2 ! 2 ! 1 !}=270$ |
| 2 identical +3 different | ${ }^{3} C _{1} \cdot{ }^{4} C _{3} \cdot \frac{5 !}{2 !}=720$ |
| All 5 different | 120 |