SMALLEST AND GREATEST NUMBER CREATED BY THE DIGITS GIVEN

What are the least and the greatest four digit numbers you can make using all the digits in each set only once ?

Problem 1 :

1, 2, 3, 4

Solution :

The given digits are 1, 2, 3, 4

The least four digit number is 1234.

The greatest four digit number 4321.

Problem 2 :

0, 3, 2, 1

Solution :

The given digits are 0, 3, 2, 1

The least four digit number is 1023

The greatest four – digit number 3210.

Problem 3 :

1, 0, 0, 2

Solution :

The given digits are 1, 0, 0, 2

The least four digit number is 1002.

The greatest four digit number 2100.

Problem 4 :

Find the number of positive integers greater than 6000 and less than 7000 which are divisible by 5, provided that no digit is to be repeated.

Solution :

Since the required number should lie in between

6000 and 7000

Available digits are 0, 1 2, 3, 4, 5, 6, 7, 8, 9

The thousand digit starts with 6 and one's digits should be 0 or 5.

 TH   H   TE   O

6  __    __    0 or 5

So far two digits is filled, still we have 8 numerical values are available.

Having 0 at the unit's place :

Number of options = 1 x 8 x 7 x 1 = 56

Having 5 at the unit's place :

Number of options = 1 x 8 x 7 x 1 = 56

So, total number of 4 digit numbers divisible by 5 is

56 + 56 = 112

112 numbers can be created which is divisible by 5.

Problem 5 :

Form the greatest and the smallest 4 digit number of the following digits:- Digits Greatest number Smallest number

a) 3,0,9,1 __________ ___________

b) 4,9,3,1 __________ ___________

c) 5,0,3,2 __________ ___________

d) 2,7,9,4 __________ ___________

Solution :

a) 3,0,9

Greatest number = 9,310

Smaller number = 1,039

b) 4,9,3,1

Greatest number = 9,431

Smallest number = 1,349

c) 5,0,3,2

Greatest number = 5,320

Smallest number = 2,035

d) 2,7,9,4

Greatest number = 9,742

Smallest number = 2,479

Problem 6 :

A five digit number divisible by 3 is to be formed using the numbers 0, 1, 2, 3, 4 and 5 without repetitions. The total number of ways this can be done is

(A) 216    (B) 600      (C) 240    (D) 3125

[Hint: 5 digit numbers can be formed using digits 0, 1, 2, 4, 5 or by using digits 1, 2, 3, 4, 5 since sum of digits in these cases is divisible by 3.]

Solution :

We have to create a five digit number.

___   ___   ____   ____   ___

In ten thousands place, we should not use 0. So, we have the remaining 5 options can be used.

If one of the digits is 0, then

Sum of (1, 2, 4 and 5) :

1 + 2 + 4 + 5 = 12

Excluding 3,

Number of 5 digit numbers = 4 x 4 x 3 x 2 x 1

= 96

If one of the digits is not 0, using the five numbers (1, 2, 3, 4, 5) we can create a 5 digit number.

Sum of (1, 2, 3, 4 and 5) :

1 + 2 + 3 + 4 + 5 = 15

Excluding 0 and including 3.

Number of 5 digit numbers = 5 x 4 x 3 x 2 x 1

= 120

Sum of two cases = 120 + 96

Number of 5 digit numbers can be formed = 216