The number of natural numbers of two or more than two digits in which digits from left to right are in increasing order is
a.127
b.128
c.502
d.501
e.512
OA: C
a.127
b.128
c.502
d.501
e.512
OA: C
This problem seems a bit too convoluted for the GMAT.parveen110 wrote:The number of natural numbers of two or more than two digits in which digits from left to right are in increasing order is
a.127
b.128
c.502
d.501
e.512
OA: C
Also, the GMAT would say positive integers rather than natural numbers.
That said, the problem does provide some worthwhile take-aways, so here's one approach:
For the digits to increase from left to right, they must be DISTINCT.
For each combination of distinct digits, there is only ONE acceptable ordering: from least to greatest.
For example, only one viable integer can be formed if the selected digits are 7, 1 and 3:
137.
Thus, for the each of the following cases, we need to count how many COMBINATIONS of distinct digits can be formed, since each combination of distinct digits will yield exactly one viable integer.
Note the following constraint:
None of the digits in the following cases can be 0.
The reason:
Since the digits must increase from left to right, any combination that includes 0 must place 0 in the leftmost position -- not possible, since the leftmost digit of a positive integer must be at least 1.
Thus, all of the following cases must be formed from the following pool of digits:
1, 2, 3, 4, 5, 6, 7, 8, 9.
We should also note the following:
nCr = nC(n-r).
To illustrate:
6C5 = 6C(6-5) = 6C1.
From 6 options, the number of ways to choose 5 = 6C5 = (6*5*4*3*2)/(5*4*3*2*1) = 6.
From 6 options, the number of ways to choose 1 = 6C1 = 6.
Thus:
6C5 = 6C1.
2-digit integers:
Number of ways to choose 2 distinct integers from 9 options = 9C2 = (9*8)/(2*1) = 36.
3-digit integers:
Number of ways to choose 3 distinct integers from 9 options = 9C3 = (9*8*7)/(3*2*1) = 84.
4-digit integers:
Number of ways to choose 4 distinct integers from 9 options = 9C4 = (9*8*7*6)/(4*3*2*1) = 126.
5-digit integers:
Number of ways to choose 5 distinct integers from 9 options = 9C5 = 9C4 = 126.
6-digit integers:
Number of ways to choose 6 distinct integers from 9 options = 9C6 = 9C3 = 84.
7-digit integers:
Number of ways to choose 7 distinct integers from 9 options = 9C7 = 9C2 = 36.
8-digit integers:
Number of ways to choose 8 distinct integers from 9 options = 9C8 = 9C1 = 9.
9-digit integers:
Here, there is only 1 option:
123456789.
Total options = 36+84+126+126+84+36+9+1 = 502.
The correct answer is C.
