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  4. There are 7 persons which include a group of 3 friends F1, F2, F3. Number of ways they can be seated on a round table if atleast two out of F 1, F2, F3 are seated next to each other is

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Q. There are $7$ persons which include a group of $3$ friends $F_{1}, F_{2}, F_{3}$. Number of ways they can be seated on a round table if atleast two out of $F _{1}$, $F_{2}, F_{3}$ are seated next to each other is

Permutations and Combinations

Solution:

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$3 ! \times{ }^{4} C_{3} \times 3 !=6 \times 4 \times 6$
$= 36 × 4 = 14$
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$\therefore $ Required ways = Total $- 144 $
$= 720 - 144 = 576$