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Circular Permutations ~ 3 Boys & 3 Girls

In how many ways can 3 boys and 3 girls be seated at a round table such that exactly 2 boys sit together?

Solution

Circular permutations without any restriction = (6 - 1)! = 120

Circular permutations when no 2 boys are together which is B G B G B G situation is a circle = 2! $\times$ 3! = 12

Circular permutations when all the 3 boys are together = (4 - 1)! $\times $ 3! = 36

Circular permutations for exactly 2 boys sitting together =  120 - (12 + 36) = 72

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$f(x)=x^6+2x^4+x^3+2x+3 $

$\mathop {\lim }\limits_{x \to 1} \frac{{{x^n}f(1) - f(x)}}{{x - 1}} = 44$

$n=?$

Let $f(x)=x^6+2x^4+x^3+2x+3,x \in R $. Then the natural number n for which $\mathop {\lim }\limits_{x \to 1} \frac{{{x^n}f(1) - f(x)}}{{x - 1}} = 44$ is _ _ _ _ . Solution Since the limit has $\left[ {\frac{0}{0}} \right]$ form, L.H. Rule is applicable. Thus, $\mathop {\lim }\limits_{x \to 1} n{x^{n - 1}}f(1) - f'(x) = 44$ $\therefore nf(1) - f'(1) = 44$ $\therefore n.9 - ({6.1^5} + {8.1^3} + {3.1^2} + 2.1) = 44$ $ \Rightarrow 9n - 19 = 44$ $\Rightarrow n=7$