Find the number of numbers, greater than a million that can be formed

1.	Find the number of numbers, greater than a million that can be formed with the digit 2, 3, 0, 3, 4, 2, 3.
Answer» Given as The digits 2, 3, 0, 3, 4, 2, 3 The total number of digits = 7 As we know that, zero cannot be the first digit of the 7 digit numbers. The number of 6 digit number = n!/ (p! × q! × r!) = 6! / (2! 3!) Ways. [2 is repeated twice and 3 is repeated 3 times] Total number of arrangements = 6! / (2! 3!) = [6 × 5 × 4 × 3 × 2 × 1] / (2 × 3 × 2) = 5 × 4 × 3 × 1 = 60 Then, number of 7 digit number = n!/ (p! × q! × r!) = 7! / (2! 3!) Ways Total number of arrangements = 7! / (2! 3!) = [7 × 6 × 5 × 4 × 3 × 2 × 1] / (2 × 3 × 2) = 7 × 5 × 4 × 3 × 1 = 420 Therefore, total numbers which is greater than 1 million = 420 – 60 = 360 Thus, total number of arrangements of 7 digits (2, 3, 0, 3, 4, 2, 3) forming a 7 digit number is 360.

Find the number of numbers, greater than a million that can be formed with the digit 2, 3, 0, 3, 4, 2, 3.

Answer»

Given as

The digits 2, 3, 0, 3, 4, 2, 3

The total number of digits = 7

As we know that, zero cannot be the first digit of the 7 digit numbers.

The number of 6 digit number = n!/ (p! × q! × r!) = 6! / (2! 3!) Ways. [2 is repeated twice and 3 is repeated 3 times]

Total number of arrangements = 6! / (2! 3!)

= [6 × 5 × 4 × 3 × 2 × 1] / (2 × 3 × 2)

= 5 × 4 × 3 × 1

= 60

Then, number of 7 digit number = n!/ (p! × q! × r!) = 7! / (2! 3!) Ways

Total number of arrangements = 7! / (2! 3!)

= [7 × 6 × 5 × 4 × 3 × 2 × 1] / (2 × 3 × 2)

= 7 × 5 × 4 × 3 × 1

= 420

Therefore, total numbers which is greater than 1 million = 420 – 60 = 360

Thus, total number of arrangements of 7 digits (2, 3, 0, 3, 4, 2, 3) forming a 7 digit number is 360.