All the natural numbers, sum of whose digits is 8 are arranged in ascending order then the `170^(th)` number in the list does not contain the digit

Options

A. 2
B. 3
C. 4
D. 5

Answer

To solve the problem of finding the `170th` number in the list of natural numbers whose digits sum up to `8`, we will follow these steps:

### Step 1: Count the numbers with digit sum equal to 8
We need to find how many natural numbers exist where the sum of the digits equals `8`. We can use the "stars and bars" theorem to count these combinations.

1. **One-digit numbers**: The only one-digit number is `8`.
- Count = 1

2. **Two-digit numbers**: We can express this as `x1 + x2 = 8`, where `x1` and `x2` are the digits of the two-digit number (with `x1` not being `0`).
- The equation becomes `x1 + x2 = 8` with `x1 >= 1` and `x2 >= 0`.
- By substituting `y1 = x1 - 1`, we have `y1 + x2 = 7` where `y1 >= 0` and `x2 >= 0`.
- The number of solutions is given by `C(7 + 1, 1) = C(8, 1) = 8`.
- Count = 8

3. **Three-digit numbers**: We can express this as `x1 + x2 + x3 = 8`, where `x1 >= 1`.
- By substituting `y1 = x1 - 1`, we have `y1 + x2 + x3 = 7`.
- The number of solutions is given by `C(7 + 2, 2) = C(9, 2) = 36`.
- Count = 36

4. **Four-digit numbers**: We express this as `x1 + x2 + x3 + x4 = 8`, where `x1 >= 1`.
- By substituting `y1 = x1 - 1`, we have `y1 + x2 + x3 + x4 = 7`.
- The number of solutions is given by `C(7 + 3, 3) = C(10, 3) = 120`.
- Count = 120

5. **Five-digit numbers**: We express this as `x1 + x2 + x3 + x4 + x5 = 8`, where `x1 >= 1`.
- By substituting `y1 = x1 - 1`, we have `y1 + x2 + x3 + x4 + x5 = 7`.
- The number of solutions is given by `C(7 + 4, 4) = C(11, 4) = 330`.
- Count = 330

### Step 2: Total Count
Now we sum all the counts:
- 1 (one-digit) + 8 (two-digit) + 36 (three-digit) + 120 (four-digit) + 330 (five-digit) = 495.

### Step 3: Finding the 170th number
Since the total count of numbers with digit sum `8` is `495`, we can find the 170th number:
- The first `1 + 8 + 36 + 120 = 165` numbers are all the one-digit, two-digit, three-digit, and four-digit numbers.
- Therefore, the `170th` number must be a five-digit number.

### Step 4: Constructing the five-digit numbers
We need to find the first few five-digit numbers whose digits sum to `8`. The smallest five-digit number is `10000`, and we can incrementally find the combinations:

1. **First five-digit number**: `10007`
2. **Next**: `10016`
3. **Next**: `10025`
4. **Next**: `10034`
5. **Next**: `10043`
6. **Next**: `10052`
7. **Next**: `10061`
8. **Next**: `10070`
9. **Next**: `10106`
10. **Next**: `10115`
11. **Next**: `10124`
12. **Next**: `10133`
13. **Next**: `10142`
14. **Next**: `10151`
15. **Next**: `10205`
16. **Next**: `10214`
17. **Next**: `10223`
18. **Next**: `10232`
19. **Next**: `10304`
20. **Next**: `10313`
21. **Next**: `10322`
22. **Next**: `10403`
23. **Next**: `10412`
24. **Next**: `10502`
25. **Next**: `11007`
26. **Next**: `11016`
27. **Next**: `11025`
28. **Next**: `11034`
29. **Next**: `11043`
30. **Next**: `11052`
31. **Next**: `11061`
32. **Next**: `11070`
33. **Next**: `11106`
34. **Next**: `11115`
35. **Next**: `11124`
36. **Next**: `11133`
37. **Next**: `11142`
38. **Next**: `11151`
39. **Next**: `11205`
40. **Next**: `11214`
41. **Next**: `11223`
42. **Next**: `11232`
43. **Next**: `11304`
44. **Next**: `11313`
45. **Next**: `11322`
46. **Next**: `11403`
47. **Next**: `11412`
48. **Next**: `11502`
49. **Next**: `12005`
50. **Next**: `12014`
51. **Next**: `12023`
52. **Next**: `12032`
53. **Next**: `12104`
54. **Next**: `12113`
55. **Next**: `12122`
56. **Next**: `12203`
57. **Next**: `12212`
58. **Next**: `12302`
59. **Next**: `13004`
60. **Next**: `13013`
61. **Next**: `13022`
62. **Next**: `13103`
63. **Next**: `13112`
64. **Next**: `13202`
65. **Next**: `14003`
66. **Next**: `14012`
67. **Next**: `14102`
68. **Next**: `15002`
69. **Next**: `20006`
70. **Next**: `20015`
71. **Next**: `20024`
72. **Next**: `20033`
73. **Next**: `20042`
74. **Next**: `20051`
75. **Next**: `20105`
76. **Next**: `20114`
77. **Next**: `20123`
78. **Next**: `20204`
79. **Next**: `20213`
80. **Next**: `20303`
81. **Next**: `20312`
82. **Next**: `20402`
83. **Next**: `21005`
84. **Next**: `21014`
85. **Next**: `21023`
86. **Next**: `21032`
87. **Next**: `21104`
88. **Next**: `21113`
89. **Next**: `21122`
90. **Next**: `21203`
91. **Next**: `21212`
92. **Next**: `21302`
93. **Next**: `22004`
94. **Next**: `22013`
95. **Next**: `22022`
96. **Next**: `22103`
97. **Next**: `22112`
98. **Next**: `22202`
99. **Next**: `23003`
100. **Next**: `23012`
101. **Next**: `23102`
102. **Next**: `24002`
103. **Next**: `30005`
104. **Next**: `30014`
105. **Next**: `30023`
106. **Next**: `30032`
107. **Next**: `30104`
108. **Next**: `30113`
109. **Next**: `30122`
110. **Next**: `30203`
111. **Next**: `30212`
112. **Next**: `30302`
113. **Next**: `31004`
114. **Next**: `31013`
115. **Next**: `31022`
116. **Next**: `31103`
117. **Next**: `31112`
118. **Next**: `31202`
119. **Next**: `32003`
120. **Next**: `32012`
121. **Next**: `32102`
122. **Next**: `33002`
123. **Next**: `40004`
124. **Next**: `40013`
125. **Next**: `40022`
126. **Next**: `40103`
127. **Next**: `40112`
128. **Next**: `40202`
129. **Next**: `41003`
130. **Next**: `41012`
131. **Next**: `41102`
132. **Next**: `42002`
133. **Next**: `50003`
134. **Next**: `50012`
135. **Next**: `50102`
136. **Next**: `51002`
137. **Next**: `60002`
138. **Next**: `70002`
139. **Next**: `80002`
140. **Next**: `90002`

After counting, we find that the `170th` number in this sequence is `10043`.

### Step 5: Identify the digits in the number
The number `10043` contains the digits `1`, `0`, `0`, `4`, and `3`.

### Conclusion
The question asks which digit is not present in the `170th` number. The digits available in the options are `2`, `3`, `4`, and `5`. Among these, `2` and `5` are not present in `10043`.

### Final Answer
The digits that do not appear in the `170th` number `10043` are `2` and `5`.