The Denton County Sheriff's Department schedules some police officers for 8-hour shifts. The beginning times for the shifts are 2:00 am, 6.00 am, 10:00 am, 2:00 pm, 6:00 pm and 10:00 pm. An officer beginning a shift at one of these times works for the next 8 hours and he/she is paid $240 for the entire shift if beginning th shift anytime from midnight to noon or $300 for the entire shift if beginning the shift anytime from noon to midnight. During normal weekday operations, the number of part-time reserve officers needed varies depending on the time of the day. The department staffing guidelines require the following minimum number o part-time reserve officers on duty (see the table below). The Denton County Sheriff's Department would like to formulate a linear program and determine the least expensive way that part-time reserve police officers could be scheduled to begin the 8-hour shifts at each of the six times (2:00 am, 6:00 am, 10:00 am, 2:00 pm, 6:00 pm and 10:00 pm). Let X- the number of officers beginning work at 2:00 am, X the number of officers beginning work at 6:00 am, etc. Time of Day Minimum Officers on Duty 2:00 am - 6:00 am 8 6:00 am -10:00 am 11 10:00 am - 2:00 pm 9 2:00 pm - 6:00 pm 7 6:00 pm - 10:00 pm 11 10:00 pm - 2:00 am 13 The constraint X+ Xg 2 11 is for: O None of the others O 6.00 pm - 10:00 pm O 10:00 am - 2:00 pm O 6:00 am - 10:00 am O 10:00 pm - 2:00 am The constraint for 10:00 am - 2:00 pm is given as: O X2+ X4 9 O X2 + X3 29 O None of the others O X3 + X49 O X1 + X229 Hiring 7 officers in each shift can be a feasible solution. O True O False

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If there are 5 dishes, they can be relished in any order at a time. In permutation, it should be in a particular order. In combination, the order does not matter. Take 3 letters a, b, and c. The possible ways of pairing any two letters are ab, bc, ac, ba, cb and ca. It is in a particular order. So, this can be called the permutation of a, b, and c. But if the order does not matter then ab is the same as ba. Similarly, bc is the same as cb and ac is the same as ca. Here the list has ab, bc, and ac alone. This can be called the combination of a, b, and c.

Counting Theory

The fundamental counting principle is a rule that is used to count the total number of possible outcomes in a given situation.