How to find every 10 digit number starting from 4?

Question

How many 10 digit numbers are there starting from number 4

There are 1 000 000 000 possible combinations of 10 digit numbers. I want to find out how many of them actually start with number 4

Rules : Find all possible combination of every 10 digit number which starts from number 4.

I am good at programming but weak at mathematics, Now when such challenges come across it becomes difficult to think about some mathematical solution. Any reference helpful would be nice. Thanks

By "starting from number 4", do you mean the first digit is 4? How many other digits are there? How many possible values are there for each? — Robert Israel, Apr 15 '15 at 23:12
Finding "every," as in the title of the question, means listing them. But the body of the question asks for a count. But then in the "rules" line, you say "find" again. — Thomas Andrews, Apr 15 '15 at 23:14
Are you using each digit $0$ through $9$ just once or will any 10 digit number starting with 4 work, like $4555411111$? — Josh B., Apr 15 '15 at 23:14
The number of 10 digit numbers is $9\times 10^9$, not $10^{10}$. — mathlove, Apr 15 '15 at 23:19
If you can count all the non-negative integers below $1000000000$, then adding $4000000000$ to each of those will give you all the $10$-digit numbers starting with $4$, — Joffan, Apr 15 '15 at 23:21

score 5 · Accepted Answer · answered Apr 15 '15 at 23:21

If your question is "how many 10digit numbers are there with the first digit being a $4$", note that all such numbers are of the form:

$$\underbrace{\underline{4}\underline{~}\underline{~}\underline{~}\underline{~}\underline{~}\underline{~}\underline{~}\underline{~}\underline{~}}_{10~\text{spaces}}$$

You can answer this directly by noting that it will be every number from $400\cdots0$ to $499\cdots 9$, or you can answer this via multiplication principle.

To see this via multiplication principle, set up the sequence of choices:

Choose the first digit. (How many choices do we have for the first digit? we required that it be a 4, so there is only one choice)
Choose the second digit. (For this, the second digit could be any of the list $0,1,2,\dots,9$ for a total of 10 possible choices)
Choose the third digit. (similarly there are 10 choices)
$\vdots$
Choose the tenth digit. (similarly there are 10 choices)

According to the multiplication principle, the total number of ways of completing the task is the product of the number of choices at each step. I.e. there are $1\cdot \underbrace{10\cdot 10\cdots 10}_{9~\text{copies}} = 1\cdot 10^9$ possible 10 digit numbers which begin with a 4.

score 0 · Answer 2 · answered Apr 15 '15 at 23:16

0

Hint: how many two digit numbers are there that start with $4$? how many three digit numbers? For three digit numbers, you have to choose two digits after the $4$. If you don't see the pattern, you could write a short program to count how many four and five digit numbers there are.

answered Apr 15 '15 at 23:16

Ross Millikan

374,822

Hey I got that pattern and it's much more efficient than writing a program to count all! , Thanks :) – Anon Dev Apr 15 '15 at 23:35
@AnonDev: This is called the rule of product If you have $a$ choices for one thing and $b$ choices for a second, you have $ab$ choices for the pair. You have ten choices per digit. – Ross Millikan Apr 15 '15 at 23:41

score 0 · Answer 3 · answered Apr 15 '15 at 23:39

Note that you can stick all ten-digit numbers into nine buckets:

$1,000,000,000$ to $1,999,999,999$
$2,000,000,000$ to $2,999,999,999$
$3,000,000,000$ to $3,999,999,999$
$4,000,000,000$ to $4,999,999,999$
$5,000,000,000$ to $5,999,999,999$
$6,000,000,000$ to $6,999,999,999$
$7,000,000,000$ to $7,999,999,999$
$8,000,000,000$ to $8,999,999,999$
$9,000,000,000$ to $9,999,999,999$

Every one of these buckets contains $1,000,000,000$ numbers.

(Don't believe me? Let's do a small example and look at the numbers between $0$ and $9$: we have $0,1,2,3,4,5,6,7,8,9$. Count them, that's $10$ of them. Do the same for $10$ through $19$: we get $10,11,12,13, \ldots ,19$ -- it's always $10$.)

All numbers starting with a $4$ are in the $4$th bucket, which contains $1,000,000,000$ numbers. So there are $1,000,000,000$ ten-digit numbers that start with a $4$.

How to find every 10 digit number starting from 4?

3 Answers3