# Chapter 06: Counting¶

## Example 01: Page 386¶

In :
n=2 #number of employees
r=12 #number of office rooms
ways_alloc_sanchez=12
ways_alloc_patel=11

#By PRODUCT RULE
print "Total ways to assign offices to these employees is",ways_alloc_sanchez*ways_alloc_patel

Total ways to assign offices to these employees is 132


## Example 02: Page 386¶

In :
alphabets=26 #Total number of alphabets
posint=100 #Total positive numbers not beyond 100

#number of chairs to be labelled with a alphabet and an integer using product rule
print "Total number of chairs that can be labelled with an alphabet and an integer is",alphabets*posint

Total number of chairs that can be labelled with an alphabet and an integer is 2600


## Example 03: Page 386¶

In :
mc=32 #total number of microcomputers
port=24 #total number of ports in each microcomputer

#total number of different ports to a microcomputer in the center are found using product rule

print "total number of ports",mc*port

total number of ports 768


## Example 04: Page 386¶

In :
bits=2 #possible bits either 0 or 1
ns=7 #number of bits in the string (ie). length of the string
# 7 bits are capable of taking either 0 or 1 so by PRODUCT RULE
print "Total different bit strings of lenth seven are",bits**ns

Total different bit strings of lenth seven are 128


## Example 05: Page 387¶

In :
letters=26 #number of letters in english alphabet
no_of_letters=3 #number of letters
choices=10 #number of choices for each letter
result=1#in order to avoid junk values.  Assigned it to 1.
for i in range(0,no_of_letters):
result=result*letters*choices
print "The total number of choices are",result

The total number of choices are 17576000


## Example 01: Page 407¶

In :
def permutation(n,r): #function definition

i=n
result=1
for i in range((n-r)+1,n+1): #computing the permutation
result=result*i

return result

print "The number of ways to select 3 students from a group of 5 students to line up for a picture is ",permutation(5,3) #function call
print "The number of ways to select 5 students from a group of 5 students to line up for a picture is ",permutation(5,5) #function call

The number of ways to select 3 students from a group of 5 students to line up for a picture is  60
The number of ways to select 5 students from a group of 5 students to line up for a picture is  120


## Example 04: Page 409¶

In :
def permutation(n,r): #function definition

i=n
result=1
for i in range((n-r)+1,n+1): #permutation computation
result=result*i
return result
num=input("Enter the number of people")
perm=input("Enter the prizes")
print "The number of ways to decide the prize winners is",permutation(num,perm) #function call

Enter the number of people100
Enter the prizes3
The number of ways to decide the prize winners is 970200


## Example 05: Page 409¶

In :
def permutation(n,r):

i=n
result=1
for i in range((n-r)+1,n+1):
result=result*i

return result
num=input("Enter the number of runners")
perm=input("Enter the number of prizes")
print "The number of ways to decide the prize winners is",permutation(num,perm)

Enter the number of runners8
Enter the number of prizes3
The number of ways to decide the prize winners is 336


## Example 06: Page 409¶

In :
def calc(n):

i=n
result=1
for i in range(1,n): #find the number of ways to decide the path. since the first city us decided. The for loop is from 1 to n
result=result*i

return result
num=input("Enter the number of cities")
print "The number of possible ways to decide the path is",calc(num)

Enter the number of cities8
The number of possible ways to decide the path is 5040


## Example 10: Page 410¶

In :
def combination(n,r): #combination function
i=n
numerator=1
denominator=1
for i in range((n-r)+1,n+1):#computes the value of the numerator
numerator=numerator*i
for j in range (1,r+1): #computes the value of the denominator
denominator=denominator*j
result=numerator/denominator #computes result
return result
num=input("Enter the number of elements")
comb=input("Enter the combinations")
print "The number of combinations are ",combination(num,comb)

Enter the number of elements4
Enter the combinations2
The number of combinations are  6


## Example 12: Page 412¶

In :
def combination(n,r): #function definition for combination
i=n
numerator=1
denominator=1
for i in range((n-r)+1,n+1):
numerator=numerator*i
for j in range (1,r+1):
denominator=denominator*j
result=numerator/denominator
return result
num=input("Enter the number of members in a team")
comb=input("Enter the number of players")
print "The number of combinations are ",combination(num,comb) #function call

Enter the number of members in a team10
Enter the number of players5
The number of combinations are  252


## Example 13: Page 412¶

In :
def combination(n,r): #function definition
i=n
numerator=1
denominator=1
for i in range((n-r)+1,n+1):
numerator=numerator*i
for j in range (1,r+1):
denominator=denominator*j
result=numerator/denominator
return result
num=input("Enter the total number of astronauts")
comb=input("Enter the number of astronauts to be selected ")
print "The total number of combinations of selected astronauts to Mars are ",combination(num,comb) #function call

Enter the total number of astronauts30
Enter the number of astronauts to be selected 6
The total number of combinations of selected astronauts to Mars are  593775


## Example 15: Page 413¶

In :
def combination(n,r): #Function definition
i=n
numerator=1
denominator=1
for i in range((n-r)+1,n+1): #computation of the numerator
numerator=numerator*i
for j in range (1,r+1): #computation of the denominator
denominator=denominator*j
result=numerator/denominator
return result
num1=input("Enter the total number of faculty in computer science department")
comb1=input("Enter the number of faculty to be selected for computer science department")
num2=input("Enter the total number of faculty in maths department")
comb2=input("Enter the number of faculty to be selected for maths department")

print "The total number of combinations of selected faculties are ",combination(num1,comb1)*combination(num2,comb2) #Function call

Enter the total number of faculty in computer science department9
Enter the number of faculty to be selected for computer science department3
Enter the total number of faculty in maths department11
Enter the number of faculty to be selected for maths department4
The total number of combinations of selected faculties are  27720