How many positive four-digit integers have their digits in ascending

First Digit - A
Second Digit - B
Third Digit - C
Fourth Digit - D

A can be 1-6
B can be 2-7
C can be 3-8
D can be 4-9
A=1; B = 2; C = 3 ; D = 6 options
A=1; B = 2; C = 4; D = 5 options
A = 1; B=2; C = 5; D = 4 options
A = 1; B=2; C = 6; D = 3 options
A = 1; B=2; C = 7; D = 2 options
A = 1; B=2; C = 8; D = 1 options
So When A =1 and B=2 then we have 21 options
WHen A = 1 and B=3 then we have 15 options
When A =1 and B=4 then we have 10 options
When A = 1 and B=5 then we have 6 options
When A = 1 and B=6 then we have 3 options
When A = 1 and B=7 then we have 1 option
So When A =1 we have 56 options
WHen A = 2 we have 35 options
When A = 3 we have 20 options
When A = 4 we have 10 options
When A = 5 we have 4 options
When A = 6 we have 1 option

Alltogether we have 56+35+20+10+4+1 = 126 options
D
Please correct me if i am wrong

Hi..

an easier, short and logical way..

choose 4 out of 10... there can be ONLY 1 way the 4 can be arranged in ascending order, so selection is what we are looking for and NOT arrangements..
\(10C4 = \frac{10!}{6!4!} = \frac{10*9*8*7}{4*3*2} = 210\)

BUT it contains numbers with 0 in thousands digit making these 3-digit numbers
so \(9C3= \frac{9!}{6!3!} = 84\)

\(ans = 210-84=126\)

D

Great explanation chetan2u

we can reduce 1 step here because \(0\) cannot be part of any number with ascending digits so we simply need to select \(4\) digits out of \(9\) digits

Hence \(9_C_4=\frac{9!}{4!5!}=126\)

Par of GMAT CLUB'S New Year's Quantitative Challenge Set

I thought of doing this question using permutation so started with filling the 1000th position and we have 9(0 can not be used at 1000th position)options for that. While doing this I realized that answer options do not have big numbers so I can easily divide by 9 and check if I find a unique answer. I divided all options by 9 and only option D passed the test. It was a shortcut to do this question.

Posted from my mobile device

Why can digits not repeat? Are 1122,2233,2344,1123 not considered valid numbers and if not, why?
Want to make sure I understand this to keep in mind this constraints for future questions

Ascending order means the digits must increase from left to right, so for example, 1122 is not in ascending order because there’s no increase from the thousands to the hundreds digit and from the tens to the units digit.