suppose 5 digit numbers are formed by digits 1 2 3 4 and 5 without repetition if we are arranged in ascending order then the hundredth number is
Answers
Answer:
51342
Step-by-step explanation:
5! = 120 nos. can be made.
Let's assume that the nos. are in ascending order.
Among the 120 nos., there are 120 / 5 = 4! = 24 nos. in which the ten thousands digits is 1, 2, 3, 4, 5 each.
This means that, among the 120 ordered nos., the ten thousands digit of first 24 nos. is 1, that of the next 24 nos. is 2, next is 3, next is 4, and next is 5. According to this, divide the nos. into 5 equal sets.
Write the possible digits in order:- 1, 2, 3, 4, 5
So, divide 100 by 24. We get quotient 4 and remainder 4.
∴ The ten thousands digit of 100th no. is 5th (quotient + 1, as remainder > 0) among the order, i.e., 5.
Remember this:- 24 × 4(quotient, as remainder > 0) = 96
So this 100th no. belongs to 5th set. Take this 5th set.
The 5th set contains only those nos. in which the ten thousands digit is 5.
In these nos. there are 24 / 4 = 3! = 6 nos. each in which thousands digit is 1, 2, 3, 4, 5 each.
This means, among the 24 nos. in this 5th set (in other sets too), the thousands digit of first 6 nos. is 1, the next 6 nos. is 2, next is 3, and next is 4. The thousands digit won't be 5 because the ten thousands digit of all these nos. is 5, and therefore, no repetition.
Write the possible digits in order:- 1, 2, 3, 4
So, according to this, divide these 24 nos. into 4 equal sets.
Okay. Subtract 24 × 4 = 96 from 100. We get 4.
Divide this 4 by 6. We get quotient 0 and remainder 4.
∴ The thousands digit of 100th no. is 1st (quotient + 1, as remainder > 0) among the order, i.e., 1.
Remember this:- 6 × 0(quotient, as remainder > 0) = 0
Okay, take this 1st set. 100th no. belongs to this set.
The thousands digit of nos. in this set is only 1.
In this set, there are 6 / 3 = 2! = 2 nos. each in which hundreds digit is 2, 3, 4 each.
Means, hundreds digit of first 2 nos. is 2, that of next 2 nos. is 3, and the next is 4. According to this, divide these 6 nos. into 3 equal sets. Digits 5 and 1 are in use.
Write in order:- 2, 3, 4.
So, subtract 6 × 0 = 0 from 4. We get 4.
Divide this 4 by 2. We get quotient 2 and remainder 0.
∴ The hundreds digit of 100th no. is 2nd (quotient, as remainder = 0) among the order, i.e., 3.
Remember this:- 2 × 1(quotient - 1, as remainder = 0) = 2
Okay, take the 2nd set. 100th no. belongs to this set.
The hundreds digit of nos. in this set is only 3.
In this set, there are 2 / 2 = 1! = 1 no. each in which tens digit is 2, 4 each.
Means, tens digit of first no. is 2 and that of next no. is 4. According to this, divide these 2 nos. into 2 equal sets. Digits 5, 1 and 3 are in use.
Write in order:- 2, 4.
So, subtract 2 × 1 = 2 from 4. We get 2.
Divide this 2 by 1. We get quotient 2 and remainder 0.
∴ The tens digit of 100th no. is 2nd (quotient, as remainder = 0) among the order, i.e., 4.
So, 5, 1, 3 and 4 are in use. So 2 becomes the ones digit of the 100th no.
∴ The 100th no. is 51342.
I was not able to explain it simply and carefully. I was also confused of this so that I took a lot of my precious time to answer this!
So if any doubts, please ask to me and I'll be able to clear them.
Thank you. Have a nice day. :-)
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