If all the six digit numbers $\mathrm{x}_{1} \mathrm{x}_{2} \mathrm{x}_{3} \mathrm{x}_{4} \mathrm{x}_{5} \mathrm{x}_{6}$ with $0<\mathrm{x}_{1}<\mathrm{x}_{2}<\mathrm{x}_{3}<\mathrm{x}_{4}<\mathrm{x}_{5}<\mathrm{x}_{6}$ are arranged in the increasing order, then the sum of the digits in the $72^{\text{th}}$ number is ____.

JEE Main · 2023 · 29 Jan 2023 · Shift 1 · Permutations and Combinations

If all the six digit numbers

x_{1} x_{2} x_{3} x_{4} x_{5} x_{6}

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

7 2^{th}

number is ____.

Answer: 32

A six-digit number with strictly increasing digits is simply a choice of six digits from

{1, 2, \dots, 9}

in their only increasing order, so there are

(6 9) = 84

of them, and listing in increasing order corresponds to increasing choices. Those beginning with

1

number

(5 8) = 56

, filling positions

1

56

; the numbers beginning with

2

follow, and among them the block

23____

gives

(4 6) = 15

more, reaching position

71

. The

72

nd number is therefore the first beginning with

24

, namely

245678

, whose digit sum is

2 + 4 + 5 + 6 + 7 + 8 = 32

JEE Main · 2023 · 29 Jan 2023 · Shift 1 · Integer answer