Difference between revisions of "2006 Alabama ARML TST Problems/Problem 4"
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==Problem== | ==Problem== | ||
Find the number of six-digit positive integers for which the digits are in increasing order. | Find the number of six-digit positive integers for which the digits are in increasing order. | ||
+ | |||
+ | NOTE: Increasing from left to right. Digits are distinct. | ||
==Solution== | ==Solution== | ||
+ | ===Solution 1=== | ||
+ | Think of a nine-digit number <math>123456789</math>. If you take out <math>3</math> digits, then it will become a <math>6</math>-digit number and all the digits will still be in increasing order. The number of ways to take three digits out is <math>\binom{9}{3}=\boxed{84}</math> | ||
+ | |||
+ | ===Solution 2=== | ||
We pick six different digits <math>a</math> through <math>f</math> for the integer. None of them can be 0, or else it is a five digit integer or the digits are not in increasing order. Let's say that <math>a</math> is the least digit of them all. <math>a</math> is therefore the hundred-thousands digit. Let's say that <math>b</math> is the second smallest integer. Then <math>b</math> is the ten-thousands digit. etc. | We pick six different digits <math>a</math> through <math>f</math> for the integer. None of them can be 0, or else it is a five digit integer or the digits are not in increasing order. Let's say that <math>a</math> is the least digit of them all. <math>a</math> is therefore the hundred-thousands digit. Let's say that <math>b</math> is the second smallest integer. Then <math>b</math> is the ten-thousands digit. etc. | ||
Latest revision as of 19:37, 12 April 2012
Problem
Find the number of six-digit positive integers for which the digits are in increasing order.
NOTE: Increasing from left to right. Digits are distinct.
Solution
Solution 1
Think of a nine-digit number . If you take out
digits, then it will become a
-digit number and all the digits will still be in increasing order. The number of ways to take three digits out is
Solution 2
We pick six different digits through
for the integer. None of them can be 0, or else it is a five digit integer or the digits are not in increasing order. Let's say that
is the least digit of them all.
is therefore the hundred-thousands digit. Let's say that
is the second smallest integer. Then
is the ten-thousands digit. etc.
For each group of we pick, there is only one arrangement such that each digit is in increasing order. There are
ways to pick the digits, therefore there are 84 integers.
See also
2006 Alabama ARML TST (Problems) | ||
Preceded by: Problem 3 |
Followed by: Problem 5 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 |