Question

How many positive integers less than 1,000,000 have the sum of their digits equal to 19?

Answer 1

Answer:

30492

Step-by-step explanation:

Hi,

Let the 6 digit number  be abcdef, here each digit could be anything

between 0 and 9,

We should have the sum of all digits of number abcdef to be 19

a + b + c + d + e + f =19

The number of non-negative integral solutions to the equation

x₁ + x₂ +.........xₐ = n is given by ⁿ⁺ᵃ⁻¹Cₐ₋₁

The number of non-negative integral solution for a, b, c ,d, e and f

are given by ²⁴C₅ = 42504  

if a=10 then b + c + d + e + f =9

The number of non-negative integral solution for a, b, c ,d, e and f

are given by ¹³C₄  = 715

if a=11 then b + c + d + e + f =8,

The number of non-negative integral solution for a, b, c ,d, e and f

are given by ¹²C₄ =495

if a=12 then b + c + d + e + f = 7

The number of non-negative integral solution for a, b, c ,d, e and f

are given by ¹¹C₄=330

if a=13 then b + c + d + e + f = 6

The number of non-negative integral solution for a, b, c ,d, e and f

are given by ¹⁰C₄ =210

if a=14 then b + c + d + e + f = 5

The number of non-negative integral solution for a, b, c ,d, e and f

are given by ⁹C₄= 126

if a=15 then b + c + d + e + f = 4

The number of non-negative integral solution for a, b, c ,d, e and f

are given by ⁸C₄ = 70

if a=16 then b + c + d + e + f = 3

The number of non-negative integral solution for a, b, c ,d, e and f

are given by ⁷C₄ = 35

if a=17 then b + c + d + e + f = 2

The number of non-negative integral solution for a, b, c ,d, e and f

are given by ⁶C₄ = 15

if a=18 then b + c + d + e + f = 1

The number of non-negative integral solution for a, b, c ,d, e and f

are given by ⁵C₄ = 5

if a=19 then b + c + d + e + f = 0,

The number of non-negative integral solution for a, b, c ,d, e and f

are given by⁴C₄ = 1  

The total number of non-negative integral solutions are

= 2002 for a>10

Since any one a,b,c ,d,e , f  can be greater than or equal to 10.

Total number of non negative integral solutions are

2002 x 6 = 12012  in which one digit ≥ 10 .

Total number of positive integers are

= 42504 - 12012 = 30492

Hope, it helps !