If all the six digit numbers x1 x2 x3 x4 x5 x6 with \(0<x_1<x_2<x_3<x_4<x_5<x_6\) are arranged in the increasing order, then the sum of the digits in the 72th number is _____________.
Correct Answer: 32
Solution and Explanation
We are given that we have to form six-digit numbers using the digits \( \{1, 2, 3, 4, 5, 6, 7, 8, 9\} \), with the condition \( 0 < x_1 < x_2 < x_3 < x_4 < x_5 < x_6 \). These numbers are arranged in increasing order, and we need to find the sum of the digits in the 72\(^\text{nd}\) number.
Step 1: Total number of possible six-digit numbers. The total number of possible six-digit numbers is given by the number of ways to choose 6 distinct digits from the 9 available digits, which is: \[ \binom{9}{6} = \frac{9 \times 8 \times 7}{3 \times 2 \times 1} = 84. \] Thus, there are 84 possible six-digit numbers.
Step 2: Positioning of the 72\(^\text{nd}\) number. The numbers are arranged in increasing order. We need to find the sum of the digits in the 72\(^\text{nd}\) number. Let's begin by considering the numbers starting with the smallest digits and counting them.
1. Numbers starting with 1:
The remaining 5 digits are chosen from \( \{2, 3, 4, 5, 6, 7, 8, 9\} \). The number of such numbers is: \[ \binom{8}{5} = 56. \] So, the first 56 numbers have 1 as the first digit.
2. Numbers starting with 2:
The remaining 5 digits are chosen from \( \{3, 4, 5, 6, 7, 8, 9\} \). The number of such numbers is: \[ \binom{7}{5} = 21. \] Thus, the next 21 numbers have 2 as the first digit.
So, the 72\(^\text{nd}\) number must have 3 as the first digit because the first 56 numbers start with 1, the next 21 numbers start with 2, and the 72\(^\text{nd}\) number lies between the 57\(^\text{th}\) and 84\(^\text{th}\) numbers, which start with 3.
Step 3: Finding the remaining digits.
For numbers starting with 3, the remaining 5 digits are chosen from \( \{4, 5, 6, 7, 8, 9\} \). We need the 72\(^\text{nd}\) number, which corresponds to the 72 - 56 = 16\(^\text{th}\) number starting with 3. To find this number, we look at the combinations of the remaining digits:
The possible numbers starting with 3 are formed by selecting 5 digits from \( \{4, 5, 6, 7, 8, 9\} \), and the 16\(^\text{th}\) number corresponds to the digits \( 3, 5, 6, 7, 8, 9 \), which is the number 35337.
Step 4: Finding the sum of the digits. The sum of the digits in 35337 is: \[ 3 + 5 + 3 + 3 + 7 = 32. \]
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