Question 1027443: In the set of three-digit integers {100,101,..., 999}, how many integers are there
a) with three distinct digits that are either increasing ( as in 257, 139) or decreasing (as in 752, 430)?
b) with three digits that are either non-decreasing ( as in 477, 555, 123) or non-increasing (666, 321, 943)? (Note that digits may repeat.)
All steps and explanations are greatly appreciated..Thanks!
Answer by Edwin McCravy(20054)   (Show Source):
You can put this solution on YOUR website!
a1. Digits in strictly ascending order:
It's 9 digits choose 3. 9C3 = 84.
That's because there is only one way to arrange
3 digits in ascending order.
It's 9C3 not 10, because 0 cannot be used at all
since the first digit must be at least 1, and
the other two digits must be greater.
Answer: 84
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a2. Digits in strictly descending order:
0 can be used, but it must come last.
Case 1. No 0's
That's the same as a1, or 84, because we can
rearrange each one in descending order.
Case 2. There is a zero last.
This is the number of two digit numbers containing
no zeros, because we can put a 0 on the right end
of each and have a three-digit number in descending
order.
The number of two digit numbers in descending
order is 9 digits choose 2 or 9C2 = 36, because for
every choice there is only one way to arrange them
in descending order. (Then we put a 0 at the end
of each).
Answer: 84+36 = 120.
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b1. Digits in non-decreasing order:
Note that we cannot use 0.
Case 1: All three digits different.
It's the 84 in the first part above.
Case 2: Those with exactly two digits alike.
We can choose the digit to have two alike in 9 ways
We can choose the digit that is not like the other
two in 8 ways. That's 9*8 = 72 ways.
Case 3: Those with all three digits the same.
They are 111,222,...,999
There are 9 of those.
Answer: 84+72+9 = 165
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b2. Digits in non-increasing order.
Case 1: No zeros.
That the same as the 165 in b1 above, for we can arrange
each one in non-increasing order. Notice that there were
no 0's.
Case 2. Exactly 1 zero, which must be at the right end.
That's the same as the 36 of case 2 of a2, arranging the
first two digits in non-increasing order, plus the 9 where
the first two can be the same: 110,220,...,990
That's 36+9=45
Case 3. Exactly 2 zeros.
That's these 9: 100,200,...900
Answer: 165+45+9 = 219
Edwin
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