Question

How many numbers greater than $6\times {{10}^{5}}$ can be formed from the digits 0, 5, 6, 7, 8 and 9 when (a) repetition not allowed (b) repetition allowed.

Answer 1

Here, we will create a general form of a 6-digit number. Now, using the general form, we will find how many numbers of ways we can fill up the first, second, third, fourth, fifth and sixth places. Now, after finding the number of ways for each place, we will multiply them with each other and find the result. The above process is used for both the sub questions.

Complete step by step answer:
We have been given $6\times {{10}^{5}}$ = 600000, now we need to form a number which is greater than the given number by using 0, 5, 6, 7, 8, 9. Therefore, we have to fill 6 numbers to make it greater than 600000.
We have,

1st	2nd	3rd	4th	5th	6th

(a) When repetition is not allowed:
For the first place, we can fill it only with 6, 7, 8, 9, hence there are only 4 ways.
For the second place, we can fill it with 0, 5, 6, 7, 8, 9 but considering one digit will be taken to fill the first place, hence there are only 5 ways.
For the third place, we can fill it with 0, 5, 6, 7, 8, 9 but the two digits are used to fill the first two places, hence there are only 4 ways.
For the fourth place, we can fill it with 0, 5, 6, 7, 8, 9 but the three digits are used to fill the first three places, hence there are only 3 ways.
Similarly, we can fill the fifth and the sixth place by 2 ways and 1 way respectively.
Total number of ways to form a number greater than 600000 $=4\times 5\times 4\times 3\times 2\times 1$
Therefore, the total number of ways to form a number greater than 600000 without repeating a single digit is 480 ways.
(b) When repetition is allowed.
Here, in the first place, we can only use 6, 7, 8, 9, hence there are only 4 ways.
For the second place, we can use 0, 5, 6, 7, 8, 9, hence there are 6 ways.
Similarly, for all the other places, there are only 6 ways.
Therefore, total number of ways to form a number greater than 600000 $=4\times 6\times 6\times 6\times 6\times 6$
$=31104$
But, if the first digit is 6 and all the other digits are 0, the number formed will not be greater than 600000, hence, we will not include that one number.
Hence, total number of ways = 31104 – 1
= 31103
Therefore, the total number of ways to form a number greater than 600000 with repetition of the digits is 31103 ways.
Hence, (a) When repetition is not allowed = 480 ways.
(b) When repetition is allowed = 31103 ways.

Note: Here, in this question, we used the fundamental principle of multiplication. It states that, “ If an operation can be performed in $m$ different ways, following a second operation can be performed in $n$ different ways, then the two operations in succession can be performed in $m\times n$ ways.