Handbook of Civil Engineering Calculations

2. Compute the number of possible assignments
   if Smith is selected
   If Smith is assigned to machine A, the number of possible assignments to B and C = 3
   x
   7
   = 21. If Smith is assigned to machine B, the number of possible assignments to A and C =
   5 x 7 = 35. if Smith is assigned to machine C, the number of possible assignments to A
   and B = 5x3 = 15. Thus, the number of possible assignments with Smith selected = 21 +
   35'+15 = 71.


3. Compute the total number of possible assignments
   By summation, the number of ways in which operators can be assigned to the three ma-
   chines = 105 + 71 = 176.

FORMATION OF PERMUTATIONS SUBJECT

TO A RESTRICTION

Permutations are to be formed of the first seven letters of the alphabet, taken four at a
time, with the restriction that d cannot be placed anywhere to the left of c. For example,
the permutation edge is unacceptable. How many permutations can be formed?

Calculation Procedure:

1. Compute the number of permutations in the absence
   of any restriction
   Use the relation Pn, = n\l(n - r)!, or P 1 ^ = 7!/3! -7x6x5x4- 840.


2. Compute the number of permutations that violate
   the imposed restriction
   Form permutations that violate the restriction. Start by placing d in the first position. Let-
   ter c can be placed in any of the three subsequent positions. Two positions now remain
   unoccupied, and five letters are available; these positions can be filled in 5 x 4 = 20 ways.
   Thus, the number of permutations in which d occupies the first position and c some sub-
   sequent position is 3 x 20 = 60. Similarly, the number of permutations in which d occu-
   pies the second position and c occupies the third or fourth position is 2 x 20 = 40, and the
   number of permutations in which d occupies the third position and c occupies the fourth
   position is 1 x 20 = 20.
   By summation, the number of unacceptable permutations = 60 + 40 + 20 = 120.


3. Compute the number of permutations that satisfy
   the requirement
   By subtraction, the number of acceptable permutations = 840 - 120 = 720.

FORMATION OF COMBINATIONS SUBJECT

TO A RESTRICTION

A committee is to consist of 6 individuals of equal rank, and 15 individuals are available
for assignment. However, McCarthy will serve only if Polanski is also on the committee.
In how many ways can the committee be formed?