Number of ways in which the letters of the word “ TOMATO” can be arranged if two alike vowels are separated, is not equal to:

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Number of ways in which letters of the word “RAITHATHA” can be arranged if no two alike letters are separated.

Number of 3 digit numbers with at least one 3 and at least one 2.

Number of ways in which Ram and Rama can exchange their maps if Ram has 3 and Rama has 7 maps, all maps being different, maintaining their original number of maps at the end.

Number of ways in which 2 alike apples and 4 alike oranges can be distributed in three children if each child one or more fruits.

answer is B, C, D.

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Detailed Solution

Answer is $| T | M | A | T |$ Number of ways $=^{5} C_{2} \times \frac{4!}{2!} = 10 \times 12 = 120$
(a) RAITHATHA
$A=3, H=2, T=2, R=1, I=1$
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$= \frac{3!}{2!} + \frac{3!}{2!} + (^{7} C_{1} \times 3!) + (2 \times 2)$
$= 3 + 3 + 42 + 4 = 52.$
(c) $M_{1}, M_{2}, ...., M_{7} | N_{1}, N_{2}, N_{3}$
Total ways = $3 (^{7} C_{1}) + 3 (^{7} C_{2}) + 1 (^{7} C_{3})$

$= 21 + 63 + 35 = 119$