Call a positive integer monotonous if it is a one-digit number or its

Question

Call a positive integer monotonous if it is a one-digit number or its digits, when read from left to right, form either a strictly increasing or a strictly decreasing sequence. For example, 3, 23578, and 987620 are monotonous, but 88, 7434, and 23557 are not. How many monotonous positive integers are there?

(A) 1024
(B) 1524
(C) 1533
(D) 1536
(E) 2048

(A) 1024
(B) 1524
(C) 1533
(D) 1536
(E) 2048

udaypratapsingh99 · Answer

divide the problem into an increasing and decreasing case:

1) Case 1: Monotonous numbers with digits in ascending order.

Arrange the digits 1 through 9 in increasing order, and exclude 0 because a positive integer cannot begin with 0.

To get a monotonous number, we can either include or exclude each of the remaining 9 digits, and there are 2^9 = 512 ways to do this. However, we cannot exclude every digit at once, so we subtract 1 to get 512-1=511 monotonous numbers for this case.

2) Case 2: Monotonous numbers with digits in descending order.

This time, we arrange all 10 digits in decreasing order and repeat the process to find 2^{10} = 1024 ways to include or exclude each digit. We cannot exclude every digit at once, and we cannot include only 0, so we subtract 2 to get 1024-2=1022 monotonous numbers for this case.

At this point, we have counted all of the single-digit monotonous numbers twice, so we must subtract 9 from our total.

Thus our final answer is 511+1022-9 =   (B) 1524

KarishmaB · Answer

Increasing Numbers: First of all, 0 cannot be a part of these numbers because it must be the first digit to be in increasing order but 0126 is the same as 126. 
So we have only 9 digits (1 - 9).
Note that we need numbers such as 129, 3789, 257 etc. To understand this, look at your number row on the keyboard. The condition is that you have to start from 1 and end at 9 and for each digit you have choice whether you press it or not going from left to right. You cannot go back.

So for each digit, you have to choice to pick it or not. Say you pick 1. Next, you don't pick 2, 3 and 4. Next you pick 5 and 6. Next you don't pick 7. Next you pick 8 and 9.
You get 15689 which will be monotonous. This will give you all 1 digit, 2 digit, 3 digit till 9 digit numbers in which the digits will be in increasing order. 
You get 2*2*2.. *2 = 2^9 distinct numbers. 
But this includes a number in which you pick nothing so you need to exclude it. 
You get 2^9 - 1 numbers.

Decreasing Numbers: Here, you can end the number with a 0 so you have 10 possible digits. Use the same logic but start with 9 this time and go on till 0.
No of numbers = 2*2*2*... 2 = 2^10 numbers.
But as before, it includes the number in which you pick nothing.
Also, it includes all 1 digit numbers again (we have already included them above) so we remove 9 numbers out of this.
Also, you need only positive integers so only 0 cannot be a monotonous number.
You get 2^10 - 1 - 9 - 1 = 2^10 - 11

Total number of numbers = 2^9 - 1 + 2^10 - 11 = 1524

Answer (B)

nick1816 · Answer

1. For ascending order case- 
No digits can be 0
We can select any number of digits from the 9 digits 1 to 9, and Possible arrangement is only 1
{ For example if we select 1,2 and 3, only way to arrange them in increasing order is 123}
Total monotonous numbers possible in this case= 9C1+9C2+9C3+.......+9C7+9C8+9C9= 2^9 - 1= 511

2. For descending order case-

We can select any number of digits from the 10 digits 0 to 9 But we don't select single digit numbers, as we have already counted them in the first case. Possible arrangement is only 1 in this case too.
Total monotonous numbers possible in this case= 10C2+10C3+.......+10C7+10C8+10C9+ 10C10 = 2^10 - 10-1= 1013

Total monotonous numbers in both cases= 1013+511= 1524

Call a positive integer monotonous if it is a one-digit number or its

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Call a positive integer monotonous if it is a one-digit number or its

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