JEE MAIN - Mathematics (2016 - 9th April Morning Slot - No. 3)
If the four letter words (need not be meaningful ) are to be formed using the
letters from the word “MEDITERRANEAN” such that the first letter is R and the fourth letter is E, then the total number of all such words is :
$${{11!} \over {{{\left( {2!} \right)}^3}}}$$
110
56
59
Explanation
Here total no of different letters present are,
(1) One M
(2) Three E (E E E)
(3) One D
(4) One I
(5) One T
(6) Two R (R R)
(7) Two A (A A)
(8) Two N (N N)
In the four letter word first letter is R and last letter is E.
$$ \therefore $$ Word is = R _ _ E
Now remaining letters are,
M, EE, D, I, T, R, AA, NN
Those 2 empty places can be filled with identical letters [EE, AA, NN] in 3 ways.
Or two empty places can be filled with distinct letters [M, E, D, I, T, R, A, N] in $${}^8{C_2} \times 2!$$ ways.
$$ \therefore $$ Total no of words = 3 + $${}^8{C_2} \times 2!$$ = 59
(1) One M
(2) Three E (E E E)
(3) One D
(4) One I
(5) One T
(6) Two R (R R)
(7) Two A (A A)
(8) Two N (N N)
In the four letter word first letter is R and last letter is E.
$$ \therefore $$ Word is = R _ _ E
Now remaining letters are,
M, EE, D, I, T, R, AA, NN
Those 2 empty places can be filled with identical letters [EE, AA, NN] in 3 ways.
Or two empty places can be filled with distinct letters [M, E, D, I, T, R, A, N] in $${}^8{C_2} \times 2!$$ ways.
$$ \therefore $$ Total no of words = 3 + $${}^8{C_2} \times 2!$$ = 59
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