How many 3 digit even numbers can be formed if repetition is not allowed?

Ex 7.3, 3 (Method 1) How many 3-digit even numbers can be made using the digits 1, 2, 3, 4, 6, 7, if no digit is repeated? We need to find 3 digit even number using 1, 2, 3, 4, 6, 7, Hence units place can have either 2, 4 or 6 Number of even numbers if 2 is at units place Hence these are 5 more digits left (1, 3, 4, 6, 7) for Hence n = 5 which we need to fill 2 place and r = 2 Number of 3 digit even number with 2 at unit place = nPr = 5P2 = 5!/((5 − 2)!) = 5!/3! = (5 × 4 × 3!)/3! = 20 Thus, Number of 3 digit even number with 2 at unit place = 20 Similarly Number of 3 digit can number with 4 at unit place = 20 and 6 at unit place = 20 Hence, Total 3-digit even numbers = 20 + 20 + 20 = 60 Ex 7.3, 3 (Method 2) How many 3-digit even numbers can be made using the digits 1, 2, 3, 4, 6, 7, if no digit is repeated? Let the 3 digit even number be Only 3 numbers are possible at units place (2 , 4 & 6) as we need even number. Number of 3 digit even numbers = 3 × 5 × 4 = 60

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Solution : No. of digits to be filled at one's place = 3
No. of digits to be filled at 10's place = 5
No. of digits to be filled at 100's place = 4
`:.` Total no. of digits formed `= 3 xx 5 xx 4 = 60`
If zero is at 100's place,
Then, no. of digits to be filled at one's place = 2
& no. of digits to be filled at 10's place = 4
`:.` No. of digits formed with zero at 100's place
`= 1 xx 2 xx 4 = 8`
`:.` Required no. of digits formed `= 60 - 8 = 52`

In mathematics, permutation is known as the process of arranging a set in which all the members of a set are arranged into some series or order. The process of permuting is known as the rearranging of its components if the set is already arranged. Permutations take place, in more or less important ways, in almost every area of mathematics. They frequently appear when different commands on certain finite sets are considered.

What is a Combination?

A combination is an act of choosing items from a group, such that (not like permutation) the order of choice does not matter. In smaller cases, it is possible to count the number of combinations. Combination refers to the union of n things taken k at a time without repetition. In combination, you can select the items in any order. To those combinations in which re-occurrence is allowed, the terms k-selection or k-combination with replication are frequently used.

Permutation Formula

In permutation r things are selected from a set of n things without any replacement. In this order of selection matter.

nPr = (n!) / (n-r)!

Here,

n = set size, the total number of items in the set

r = subset size , the number of items to be selected from the set

Combination Formula

In combination r things are selected from a set of n things and where the order of selection does not matter.

nCr = n!/(n−r)!r!

Here, 

n = Number of items in set

r = Number of items selected from the set

How many 3-digit even numbers can be formed by using the digits 1,2,3,4, and 5?

Solution:

If repetition is allowed  

A three digit even number is to be formed from given 5 digits 1,2,3,4,5.

Ones place can be filled by 2 or 4 since the number is to be even. So, there are 2 ways to fill ones place.

The number system is a system of processing numbers. Number formats are the internal representation of numeric values in computer architecture. Numeric values are stored as groupings of bits, such as bytes and words. There are four different types of Number systems, the decimal system is mostly used, and the Binary system is used by computers. 

The four different types of number systems,

1. Binary number system

This number system uses only two digits 0 and 1 followed with base 2. Binary digits also known as bits. Computers understand the Binary system only.

2. Octal number system

The octal number system referred to a base 8. It contains eight symbols.i.e;  0, 1, 2, 3, 4, 5, 6, 7.

3. Decimal number system

 It is the number system that uses digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 followed with base 10. In mathematics, decimal system is generally used.

4. Hexadecimal number system

Sixteen symbols used in hexadecimal system are 0,1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F.

A bit in Binary number

A single binary digit is called a bit. The binary number is made up of only 0’s and 1’s. These are the only two digits used and their combination in different ways is used to form different values/numbers. 

Example:

101 is a three-bit binary number

10100 is a five-bit binary number

How many values can be represented in 3 digit binary?

A three-digit binary simply means the arrangement of three 0’s and 1’s in all possible manners to form different values, for instance, 000, 001, 011, 100, etc. Since there are three digits, the maximum number of ways they can be arranged is 23 which is equal to 8. Hence, it gives numbers from 0-7. Below is the way to identify the number based on the three-digit binary, the unit digit represents 20x , the tens digit represents 21x, the hundredth base represents 22x, where x= 0 or 1.

How many even numbers of 3 digits can be formed if repetition is allowed?

As the repetition is allowed, So the number of digits available for X = 6, Similarly, the number of digits available for Y = 6. Thus, The total number of 3-digit even numbers that can be formed = 6×6×3 = 108.

How many 3

Required number of numbers =(9×9×8)=648.

How many 3

so , ans=3*5*4=60. Q.

How many even numbers of 3 different digits can be formed?

The number will be even if last digit is 2, 4, 6 or 8 i.e., the last digit can be filled in 4 ways and remaining two digits can be filled in 8P2 ways. Hence required number of numbers are 8P2×4=224.