The number of 4 digit numbers having their digits in non-decreasing order…
2015
The number of 4 digit numbers having their digits in non-decreasing order (from left to right) constructed by using the digits belonging to the set {1, 2, 3} is____________.
Attempted by 36 students.
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Correct answer: 15
Solution: Count the number of 4-digit sequences using digits 1, 2, 3 that are non-decreasing from left to right.
Think in terms of how many times each digit appears. Let x1, x2, x3 be the counts of digits 1, 2, 3 respectively. These counts satisfy x1 + x2 + x3 = 4 with each xi ≥ 0.
The number of nonnegative integer solutions to x1 + x2 + x3 = 4 is given by combinations with repetition: C(4 + 3 - 1, 3 - 1) = C(6, 2) = 15.
Each solution corresponds uniquely to a non-decreasing 4-digit number, so the total count is 15.
Answer: 15