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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

### A man starts walking from the point P (-3, 4) ....

A man starts walking from the point P (-3, 4), touches the x-axis at R, and then turns to reach at the point Q (0, 2). The man is walking at a constant speed. If the man reaches the point Q in the minimum time, then $50 [(PR)^2 + (RQ)^2 ]$ is equal to _ _ _ _ . Solution For time to be minimum at constant speed, the directions must be symmetric. In other words, the angles made by PR and RQ with the vertical must be the same just like in the law of reflection in optics. $tan \theta = \frac {MP}{MR} = \frac {NQ}{NR}$ $\Rightarrow \frac {3-r}{4} = \frac {r}{2}$ $\Rightarrow r=1$ So, $R \equiv ( - 1,0)$ Now, $50(PR^2+RQ^2)=50[(4+16)+(1+4)]=1250$