How many different words can be formed (with or without meaning) with the letters of the word $ALGEBRA$? How many of these begin with $L$ and end with $R$?
The word $\;$ $ALGEBRA$ $\;$ has $7$ letters of which two are $A's$ and the rest are different.
$\therefore \;$ Number of words which can be formed with the letters of the word $ALGEBRA$ are
$= \dfrac{7!}{2!} = \dfrac{7 \times 6 \times 5 \times 4 \times 3 \times 2!}{2!} = 2520$
When the words begin with $L$ and end with $R$, the first and the seventh place can be selected in $1$ way each.
The remaining five places can be selected from the letters $A, \; G, \; E, \; B, \; A$ in
$\dfrac{5!}{2!} = \dfrac{5 \times 4 \times 3 \times 2!}{2!} = 60$ ways
$\therefore \;$ Number of words which begin with $L$ and end with $R$ $= 1 \times 60 \times 1 = 60$ words