# Formation of Numbers with the Given Digits

In formation of numbers with the given digits we may say that a number is an arranged group of digits. Numbers may be formed with or without the repetition of digits. Let us observe some of the formation of numbers without the repetition of digits.

Some 2-digit numbers formed by given digits are shown below:

(i) with digits 2 and 3 the formed numbers are 23, 32

(ii) with digits 4 and 7 we may form the numbers 47, 74

(iii) with digits 5 and 9 we may form the numbers 59, 95

But with the digit 0 and the digit 6, only the number 60 is formed. The number 06 is not a two-digit number.

Accordingly, 3-digit numbers are also formed.

Some numbers formed by 3 digits are given below:

(a) The numbers by 2, 3, and 4 are:

234, 243, 324, 342, 423, 432 (6 numbers each of 3 digits)

(b) The numbers by 4, 6 and 9 are:

469, 496, 649, 694, 946, 964 (6 numbers each of 3 digits)

(c) The numbers by 3, 7 and 8 are:

378, 387, 738, 783, 837, 873 (6 numbers each of 3 digits)

(d) The numbers by 0, 1 and 5 are

105, 150, 501, 510 (only 4 numbers of 3 digits)

Some numbers formed by 4 digits are given below:

(a) The numbers formed by 1, 2, 3 and 4 are:

1234, 1243, 1324, 1342, 1423, 1432

2134, 2143, 2314, 2341, 2413, 2431

3124, 3142, 3421, 3412, 3214, 3241

4123, 4132, 4213, 4231, 4312, 4321

(b) Similarly, other 4-digit numbers may be formed.

We observe that, when the numbers of digits are increased, more and more numbers are formed. We are familiar with the formation of numbers with 2 or 3 digits. We should know the procedure of forming the greatest and smallest numbers having definite digits.

Examples on Formation of Numbers with the Given Digits:

1. How many 3 digit numbers can be formed by using the digits 0, 3 and 7 without repeating any digit.

Solution:

Placing 0 at one's place, we can have two natural numbers 730 and 370.

While placing 0 at tens place we can have two natural numbers 307 and 703.

Zero cannot be placed at hundreds place. Hence, the numbers are 370, 730, 307 and 703.

2. The greatest 6-digit number formed by using the digits 9, 7, 0, 5 (where two of the four digits are allowed to repeat twice only) is

(a) 779950

(b) 999750

(c) 997750

(d) 997500

3. Form the greatest 9-digit number having six different digits is __________.

(a) 999998656

(b) 999999999

(c) 998877663

(d) 999987654

4. Without repeating any digit, write the smallest and largest possible 4-digit numbers having 6 in tens place.

Smallest possible 4-digit numbers having 6 in tens place is 1062

Largest possible 4-digit numbers having 6 in tens place is 9867

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