Six children â€” A, B, C, D, E, and F â€” are going to sit in six chairs in a row. Child E must be somewhere to the left of child F. How many possible configurations are there for the children?  Choose the correct choice:  A) 60  B) 180  C) 240  D) 360  E) 720
If we wanted, we could make this one extremely difficult, counting out all kinds of possibilities in several different cases. Instead, we are going to make this ridiculously easy.   First of all, with absolutely no restrictions, how many ways can the six children be arranged on the six chairs? Thatâ€™s a permutation of the 6 items â€”- 6P6 = 6! = 720. Thatâ€™s the total number of arrangements with no restrictions. Of course, those 720 arrangements have all kinds of symmetry to them. In particular, in all of those arrangements overall, itâ€™s just as likely for E to be to the left of F as it is for E to be to the right of F. Therefore, exactly half must have E to the right of F, and exactly half must have E to the left of F. Therefore, exactly (1/2)*720= 
<gadget id="calculator">(1/2)*720</gadget>
<output>360</output>
360 of the arrangements have E to the left of F. Answer= 360.
<result>D</result>
Six children â€” A, B, C, D, E, and F â€” are going to sit in six chairs in a row. Child E must be somewhere to the left of child F. How many possible configurations are there for the children?