A machine is being designed that will contain seven differen

A machine is being designed that will contain seven different controls. These controls designed C1,C2,C3,C4,C5,C6, and C7 and laid out in a linear arrangement. In how many different ways can controls C5,C6 and C7 be arranged suppose controls C1,C3,C4 and C7 need to be grouped together (i.e., they need be adjacent to one another),but in no particular order Given this constraint, in how many different ways can these 7 controls be arranged? suppose C1,C2 and C3 are used for emergency operations and controls C5,C6, and C7 are used for normal operations. If four controls are randomly select without replacement from these seven controls, what what is the probability of selecting urgency control and three normal ??ntrols?

Solution

Controls are C1,C2,C3,C4,C5,C6,C7;
It can be arranged in 7! ways
a)
No. of ways C5,C6,C7 are to be arranged are = 3! = 6 ways
b)
When C1,C3,C4,C7 need to be grouped together;
Joing this 4 elements in to one group + Rest are 3 = We have 4 elements
Arrangement of 4 diffrent elements = 4! & we know that 4 elements are group together , this coulb be also arranged in 4! ways.
Finally arrangements are = 4! * 4! = 24*24 = 576 Ways
c)
Selecting 1 emergency Control & 3 normal controls = 4 C 3 * 3 C 1 / 7 C 4 = (4 * 3) (4*3*2*1)/ 7*6*5*4 = 0.3428