Question

How many natural numbers not exceeding 4321 can be formed with the digits 1,2,3 and 4,if the digits can repeat?

The answer is 229.I just need the steps.

Answer 1

$\huge{Hey Dude!!!}$

☆☞ Here is ur answer ☜☆

☆☞ The given digits are 1, 2, 3 and 4. These digits can be repeated while forming the numbers. So, number of required four digit natural numbers can be found as

☆☞ Consider four digit natural numbers whose digit at thousandths place is

☆☞ 1. Here, hundredths place can be filled in 4 ways.   (Using the digits 1 or 2 or 3 or 4)

☆☞Similarly, tens place can be filled in 4 ways.  (Using the digits 1 or 2 or 3 or 4)

☆☞ Ones place can be filled in 4 ways.    (Using the digits 1 or 2 or 3 or 4)

☆☞ Number of four digit natural numbers whose digit at thousandths place is 1 = 4 × 4 × 4 = 64

☆☞ Similarly, number of four digit natural numbers whose digit at thousandths place is 2 = 4 × 4 × 4 = 64

☆☞ Number of four digit natural numbers whose digit at thousandths place is 3 = 4 × 4 × 4 = 64

☆☞ Now, consider four digit natural numbers whose digit at thousandths place is 4:

☆☞ Here, if the digit at hundredths place is 1, then tens place can be filled in 4 ways and ones place can also be filled in 4 ways.

☆☞ If the digit at hundredths place is 2, then tens place can be filled in 4 ways and ones place can also be filled in 4 ways.

☆☞ If the digit at hundredths place is 3 and the digit at tens place is 1, then ones place can be filled in 4 ways.

☆☞ If the digit at hundredths place is 3 and the digit at tens place is 2, then ones place can be filled only in 1 way so that the number formed is not exceeding 4321.

☆☞ Number of four digit natural numbers not exceeding 4321 and digit at thousandths place is 3 = 4 × 4 + 4 × 4 + 4 + 1 = 37

☆☞ Thus, required number of four digit natural numbers not exceeding 4321 is 64 + 64 + 64 + 37 = 229.

Enjoy....

Good bye!!!

Answer 2

heya...

Here is your answer...

No. has to be less than 4321

so, consider no. s less than 4000

 

1st place:3 ways

2nd place:4 ways

3rd palce:4 ways

4th place:4 ways

therefore, 3*4*4*4=192

 

now consider no.s starting with 4

2nd place can be filled in 2 ways (1&2) as no. has to be less than 4321

1st place:1 way

2nd place:2 ways

3rd palce:4 ways

4th place:4 ways

therefore,1*2*4*4=32

 

now consider no.s starting with 4 and followed by 3

case 1: 432_ → 1 way

case 2:431_ → 4 ways

therfore , 1+4= 5 ways

 

So , total no. of numbers formed= 192+32+5=229


It may help you...☺☺