Question 252429: Principles of Counting or Permutations. What formula do I use here.
Telephone Numbers. How many 7-digit telephone numbers are possible if the first digit cannot be 0 and:
a.) Only odd digits may be used. Answer: (78,125)
b.) The telephone number must be a multiple of 10 (that is, it must end in 0). Answer:(900,000)
c.) The telephone number must be a multiple of 100. Answer:(90,000)
d.) The first 3 digits are 481. Answer:(10,000)
e.) No repetitions are allowed. Answer:(544,320)
Answer by palanisamy(496)   (Show Source):
You can put this solution on YOUR website! We have to form 7 digit telephone numbers.
a)The odd numbers are 1,3,5,7,9
Every digit can be filled with any one of these numbers in 5 ways.
So the number of 7 digit numbers = 5*5*5*5*5*5*5
= 78,125
b)The telephone number must be a multiple of 10
So, the noumber must end with 0. Fill the unit place with 0.
This can be done in only one way.
The number cannot begin with 0.
So the first place can be filled with any one of the numbers
1,2,3,4,5,6,7,8,9
This can be done in 9 ways.
The second place can be filled with any one of the numbers
0,1,2,3,4,5,6,7,8,9
This can be done in 10 ways.
The third place can be filled with any one of the numbers
0,1,2,3,4,5,6,7,8,9
This can be done in 10 ways.
The fourth place can be filled with any one of the numbers
0,1,2,3,4,5,6,7,8,9
This can be done in 10 ways.
The fifth place can be filled with any one of the numbers
0,1,2,3,4,5,6,7,8,9
This can be done in 10 ways.
The sixth place can be filled with any one of the numbers
0,1,2,3,4,5,6,7,8,9
This can be done in 10 ways.
Therefore the total number of 7 digit telephone numbers
which are multiples of 10 = 1*9*10*10*10*10*10
= 900,000
c)The telephone number must be a multiple of 100
Fill the unit place with 0 and the tenth place with 0.
This can be done in only one way.
The number cannot begin with 0.
So the first place can be filled with any one of the numbers
1,2,3,4,5,6,7,8,9
This can be done in 9 ways.
The second place can be filled with any one of the numbers
0,1,2,3,4,5,6,7,8,9
This can be done in 10 ways.
The third place can be filled with any one of the numbers
0,1,2,3,4,5,6,7,8,9
This can be done in 10 ways.
The fourth place can be filled with any one of the numbers
0,1,2,3,4,5,6,7,8,9
This can be done in 10 ways.
The fifth place can be filled with any one of the numbers
0,1,2,3,4,5,6,7,8,9
This can be done in 10 ways.
Therefore the total number of 7 digit telephone numbers
which are multiples of 100 = 1*9*10*10*10*10*
= 90,000
d)Fill the first three digits as 481.
This can be done in only one way.
The fourth place can be filled with any one of the numbers
0,1,2,3,4,5,6,7,8,9
This can be done in 10 ways.
The fifth place can be filled with any one of the numbers
0,1,2,3,4,5,6,7,8,9
This can be done in 10 ways.
The sixth place can be filled with any one of the numbers
0,1,2,3,4,5,6,7,8,9
This can be done in 10 ways.
The seventh place can be filled with any one of the numbers
0,1,2,3,4,5,6,7,8,9
This can be done in 10 ways.
Therefore the total number of 7 digit telephone numbers
which begin with 481 are = 10*10*10*10*
= 10,000
e) No repetitions are allowed.
The number cannot begin with 0.
So the first place can be filled with any one of the numbers
1,2,3,4,5,6,7,8,9
This can be done in 9 ways.
After filling the first place we have 9 numbers and so
the second place can be filled with any one of the numbersin 9 ways.
After filling the second place we have 8 numbers and so
the third place can be filled with any one of the numbersin 8 ways.
After filling the third place we have 7 numbers and so
the foutrh place can be filled with any one of the numbersin 7 ways.
After filling the fourth place we have 6 numbers and so
the fifth place can be filled with any one of the numbersin 6 ways.
After filling the fifth place we have 5 numbers and so
the sixth place can be filled with any one of the numbersin 5 ways.
After filling the sixth place we have 4 numbers and so
the seventh place can be filled with any one of the numbersin 4 ways.
Therefore the total number of 7 digit telephone numbers
which are formed without repetition =9*9*8*7*6*5*4
= 544,320
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