How many odd numbers with distinct digits can be created using the digits 1, 2,3,4,5,6

How many numbers with distinct digits are there between 1000 and 9999. [1]

I came up with a solution like this.

Since we can't know what numbers have been used, in the tens, hundreds and thousands we start counting at the ones.

1s:     { 1, 3, 5, 7, 9 }, so 5 initial possibilities
10s:    { 0, 1, ... , 9 }, so 10 initial possibilities, 1 taken: 9 left
100s:   { 0, 1, ... , 9 }, so 10 initial possibilities, 2 taken: 8 left
1000s:  { 1, 2, ... , 9 }, so 9 initial possibilites, 3 taken: 6 left

So then we arrive at the following: 5 * 9 * 8 * 6 = 2160 possibilities. I thought this was pretty straight forward.

Than I had a glimpse at the solution sheet... And lo an answer which really doesn't make much sense at its first glimpse.

Calculate the sum of those odd numbers with distinct digits with no 0’s, a 0 in the tens place, or a 0 in the hundreds place. No 0’s: 5 choices for the ones place, then 8 · 7 · 6 choices for the other three places; 0 in the tens place: 5 choices for the ones place and 1 choice for the tens place, then 8 · 7 choices for the other two places; 0 in the hundreds place: 5 choices for the ones place and 1 choice for the hundreds place, then 8 · 7 choices for the other two places;

(5 · 8 · 7 · 6) + (5 · 1 · 8 · 7) + (5 · 1 · 8 · 7) = 2240;

Why are the 0's treated special? The exercise states it should be an odd number, with distinct digits. I thought I adhered to that proposition....

[1] Exercise 2.7.15 from Applied Combinatorics 2nd edition by Fred S. Roberts and Barry Tesman

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

No worries! We‘ve got your back. Try BYJU‘S free classes today!

No worries! We‘ve got your back. Try BYJU‘S free classes today!

No worries! We‘ve got your back. Try BYJU‘S free classes today!

No worries! We‘ve got your back. Try BYJU‘S free classes today!

Solution

The correct option is B (i) 60, (ii) 108(i) When repetition of digits is not allowed: Since we have to form a 3-digit odd number, thus the digit at unit's place must be odd. Hence, the unit's place can be filled up by 1, 3 or 5, that is, in 3 ways. Now, the ten's digit can be filled up by any of the remaining 5 digits in 5 ways and then the hundred's place can be filled up by the remaining 4 digits in 4 ways. Hence, the number of 3-digit odd numbers that can be formed =3×4×5=60. (ii) When repetition of digits is allowed: Again, the unit's place can be filled up by 1, 3, 5. that is, in 3 ways. But the ten's and hundred's place can be filled up by any of the 6 given digits in 6 ways each. (since repetition is allowed) Hence, the number of 3-digit odd numbers that can be formed =3×6×6=108.

Nội dung chính Show

• The correct option is B (i) 60, (ii) 108(i) When repetition of digits is not allowed: Since we have to form a 3-digit odd number, thus the digit at unit's place must be odd. Hence, the unit's place can be filled up by 1, 3 or 5, that is, in 3 ways. Now, the ten's digit can be filled up by any of the remaining 5 digits in 5 ways and then the hundred's place can be filled up by the remaining 4 digits in 4 ways. Hence, the number of 3-digit odd numbers that can be formed =3×4×5=60. (ii) When repetition of digits is allowed: Again, the unit's place can be filled up by 1, 3, 5. that is, in 3 ways. But the ten's and hundred's place can be filled up by any of the 6 given digits in 6 ways each. (since repetition is allowed) Hence, the number of 3-digit odd numbers that can be formed =3×6×6=108.
• How many 3-digit even numbers can be formed by using the digits 1,2,3,4, and 5?
• Similar Questions
• How many 5 digit even numbers with distinct digits can be formed using the digits 3 4 7 78?
• How many 5 digit even numbers with distinct digits can be formed using the digits?
• How many 5 digit numbers are there with distinct digits can be formed using the digits 1,2 5 5 4?
• How many different 5 digit even numbers can be formed?
• How many odd numbers of 5 digits can be formed using all the digits from 0 to 9 if repetition of digits is allowed?
• How many 5 digit even numbers with distinct digits can be formed using the digits?
• How many number of five digits can be formed with the digits 0 1,2 3 4 without repeating a digit?
• How many 5 digit numbers can be formed with exactly two distinct digits?
• How many 3 digit odd numbers with distinct digits are there?
• How many 3 digit odd numbers can be formed using the digits 0 1 2 3 4 and 5 if I repetition of digits is allowed II Repeation of digits are not allowed?
• How many three digit odd numbers can be formed using the digits 1 2 4 and 6 if repetition of digits is not allowed?
• How many 3 digit numbers can be formed from the digits 1 2 3 4 and 5 how many different 3 digit odd numbers can be formed?

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

No worries! We‘ve got your back. Try BYJU‘S free classes today!

No worries! We‘ve got your back. Try BYJU‘S free classes today!

No worries! We‘ve got your back. Try BYJU‘S free classes today!

No worries! We‘ve got your back. Try BYJU‘S free classes today!

Solution

The correct option is B (i) 60, (ii) 108(i) When repetition of digits is not allowed: Since we have to form a 3-digit odd number, thus the digit at unit's place must be odd. Hence, the unit's place can be filled up by 1, 3 or 5, that is, in 3 ways. Now, the ten's digit can be filled up by any of the remaining 5 digits in 5 ways and then the hundred's place can be filled up by the remaining 4 digits in 4 ways. Hence, the number of 3-digit odd numbers that can be formed =3×4×5=60. (ii) When repetition of digits is allowed: Again, the unit's place can be filled up by 1, 3, 5. that is, in 3 ways. But the ten's and hundred's place can be filled up by any of the 6 given digits in 6 ways each. (since repetition is allowed) Hence, the number of 3-digit odd numbers that can be formed =3×6×6=108.

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.

Nội dung chính Show

• How many 3-digit even numbers can be formed by using the digits 1,2,3,4, and 5?
• Similar Questions
• How many 5 digit even numbers with distinct digits can be formed using the digits 3 4 7 78?
• How many 5 digit even numbers with distinct digits can be formed using the digits?
• How many 5 digit numbers are there with distinct digits can be formed using the digits 1,2 5 5 4?
• How many different 5 digit even numbers can be formed?
• How many odd numbers of 5 digits can be formed using all the digits from 0 to 9 if repetition of digits is allowed?
• How many 5 digit even numbers with distinct digits can be formed using the digits?
• How many number of five digits can be formed with the digits 0 1,2 3 4 without repeating a digit?
• How many 5 digit numbers can be formed with exactly two distinct digits?

Nội dung chính Show

• How many 3-digit even numbers can be formed by using the digits 1,2,3,4, and 5?
• Similar Questions
• How many 5 digit even numbers with distinct digits can be formed using the digits 3 4 7 78?
• How many 5 digit even numbers with distinct digits can be formed using the digits?
• How many 5 digit numbers are there with distinct digits can be formed using the digits 1,2 5 5 4?
• How many different 5 digit even numbers can be formed?

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.

Since, repetition is allowed , so tens place can be filled by 5 ways.

Likewise, hundreds place can also be filled by 5 ways.

So, number of ways in which three digit even numbers can be formed is 5 × 5 × 2 = 50

If repetition is not 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.

Since, repetition is not allowed, so tens place can be filled by 4 ways.

Similarly, hundreds place can be filled by 3 ways.

So, number of ways in which three digit even numbers can be formed is 2 × 4 × 3 = 24

Similar Questions

Question 1: How many 3 digit odd numbers can be formed by using the digits 1,2,3,4 and 5?

Solution:

If repetition is allowed  

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

Ones place can be filled by 1, 3 or 5 since the number is to be odd. So,

there are 3 ways to fill ones place.

Since, repetition is allowed , so tens place can be filled by 5 ways.

Similarly, hundreds place can also be filled by 5 ways.

So, number of ways in which three digit odd numbers can be formed is 5×5×3=75

If repetition is not allowed

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

Since, for the number is to be odd , so ones place can be filled by 1, 3 or 5. So,

there are 3 ways to fill ones place.

Since, repetition is not allowed , so tens place can  be filled by 4 ways.

Similarly, hundreds place can  be filled by 3 ways.

So, number of ways in which three digit odd numbers can be formed is 3×4×3 =36

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

Solution:

If repetition is allowed  

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

Since, for the number is to be even, so ones place can be filled by 2 or 4. So, there

are 2 ways to fill ones place.

Since, repetition is allowed, so tens place can be filled by 5 ways.

Similarly, hundreds place can also be filled by 5 ways.

Similarly, thousandth place can also be filled by 5 ways

So, number of ways in which four digit even numbers can be formed is 5 × 5 × 5 × 2 = 250

If repetition is not allowed

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

Since, for the number is to be even, so ones place can be filled by 2 or 4. So,

there are 2 ways to fill ones place.

Since, repetition is not allowed, so tens place can be filled by 4 ways.

Similarly, hundreds place can be filled by 3 ways.

Similarly, thousandth place can be filled by 2 ways

So, number of ways in which four digit even numbers can be formed is 2 × 4 × 3 × 2 = 48

How many 5 digit even numbers with distinct digits can be formed using the digits 3 4 7 78?

Therefore, there will be 60 distinct 5 - digit numbers. Was this answer helpful?

How many 5 digit even numbers with distinct digits can be formed using the digits?

Case 1: If 0 is filled in units place then the number of ways = 5 × 6 × 6 × 6 × 1 = 1,080 ways. Case 2: If 0 is not filled in units place then the number of ways = 5 × 6 × 6 × 6 × 2 = 10 × 216 = 2,160. Total number of ways = 1,080 + 2,160 = 3,240.

How many 5 digit numbers are there with distinct digits can be formed using the digits 1,2 5 5 4?

Answer. Step-by-step explanation: 36 is the digital even number with distinct digit can be formed using the digit 1,2,5,5,4.

How many different 5 digit even numbers can be formed?

The question was, how many 5 digit even numbers can be made if repetition is not allowed and the number four must be included. the answer is 7686.

How many odd numbers of 5 digits can be formed using all the digits from 0 to 9 if repetition of digits is allowed?

∴ The required result will be 192.

How many 5 digit even numbers with distinct digits can be formed using the digits?

Answer & Solution The 5 digit even numbers can be formed out of 1, 2, 5, 5, 4 by using either 2 or 4 in the unit's place. This can be done in 2 ways. Corresponding to each such arrangement, the remaining 4 places can be filled up by any of the remaining four digits in 4! / 2! = 12 ways.

How many number of five digits can be formed with the digits 0 1,2 3 4 without repeating a digit?

∴ Total number of 5–digit numbers that can be formed using the digits 0, 1, 2, 3, 4 = 5! – 4! = 120 – 24 = 96.

How many 5 digit numbers can be formed with exactly two distinct digits?

The total is 9⋅9⋅15=1215.

How many 3 digit odd numbers with distinct digits are there?

Thus there are 60, 3 digit numbers, with distinct digits, with each of the digits odd. Note: We should be very careful about multiplying those numbers.

How many 3 digit odd numbers can be formed using the digits 0 1 2 3 4 and 5 if I repetition of digits is allowed II Repeation of digits are not allowed?

Hence, the number of 3-digit odd numbers that can be formed =3×6×6=108.

How many three digit odd numbers can be formed using the digits 1 2 4 and 6 if repetition of digits is not allowed?

But the ten's and hundred's place can be filled up by any of the 6 given digits in 6 ways each.
Hence, the number of 3-digit odd numbers that can be formed =`3xx6xx6=108.

How many 3 digit numbers can be formed from the digits 1 2 3 4 and 5 how many different 3 digit odd numbers can be formed?

How many 3-digit numbers can be formed from the digits 1, 2, 3, 4 and 5 assuming that repetition of the digits is not allowed. = m x n x p = 5 x 4 x 3 = 60.

How many numbers with distinct digits can be formed using some or digits 2 3 5 7 9?

How many odd numbers does 5 distinct have?

How many 3

How many distinct numbers are there with 5 digits?