Q-how-many-natural-numbers-less-than-6000-exist-such-as-the-sum-of-their-digits-equal-to-8-choices-155-165-164-158-

Question Number 172802 by mnjuly1970 last updated on 01/Jul/22

Q : how many natural numbers less than

6000, exist such as the sum of

their digits equal to 8 ?

choices : 155 165 164 158

Answered by mr W last updated on 02/Jul/22

\underset{―}{1 - d i g i t n u m b e r s :}

8

\Rightarrow 1 n u m b e r

\underset{―}{2 - d i g i t n u m b e r s :}

17 / 71, 26 / 62, 35 / 53, 44, 80

\Rightarrow 8 n u m b e r s

\underset{―}{3 - d i g i t n u m b e r s :}

a b c w i t h a + b + c = 8 a n d

a ⩾ 1 a n d b, c ⩾ 0

(x + x^{2} + \dots) {(1 + x + x^{2} + \dots)}^{2} = x \underset{k = 0}{\sum^{\infty}} C_{2}^{k + 2} x^{k}

\Rightarrow C_{2}^{7 + 2} = 36 n u m b e r s

\underset{―}{4 - d i g i t n u m b e r s :}

a b c d w i t h a + b + c + d = 8 a n d

1 ⩽ a ⩽ 5 a n d b, c, d ⩾ 0

(x + x^{2} + x^{3} + x^{4} + x^{5}) {(1 + x + x^{2} + \dots)}^{3}

= x (1 - x^{5}) \underset{k = 0}{\sum^{\infty}} C_{3}^{k + 3} x^{k}

\Rightarrow C_{3}^{7 + 3} - C_{3}^{2 + 3} = 110 n u m b e r s

\underset{―}{t o t a l l y :}

1 + 8 + 36 + 110 = 155 n u m b e r s e x i s t . ✓

o r

(1 + x + x^{2} + x^{3} + x^{4} + x^{5}) {(1 + x + x^{2} + \dots)}^{3}

= (1 - x^{6}) \underset{k = 0}{\sum^{\infty}} C_{3}^{k + 3} x^{k}

\Rightarrow C_{3}^{8 + 3} - C_{3}^{2 + 3} = 155 n u m b e r s

Commented by Tawa11 last updated on 01/Jul/22

Great sir

Commented by mr W last updated on 02/Jul/22

y o u m e a n t h e a n s w e r i s c o r r e c t M i s s ?

Commented by mahdipoor last updated on 02/Jul/22

a b c d w i t h a + b + c + d = 8 a n d a, b, c, d ⩾ 0

\Rightarrow C_{k - 1}^{n + k - 1} = C_{3}^{11} = 165

a b c d w i t h a + b + c + d = 8 a n d a, b, c, d ⩾ 0

a n d b i g g e r t h a n 6000 :

{\begin{cases} a = 6 \Rightarrow b + c + d = 2 \Rightarrow C_{2}^{4} = 6 \\ a = 7 \Rightarrow b + c + d = 1 \Rightarrow C_{2}^{3} = 3 \\ a = 8 \Rightarrow b + c + d = 0 \Rightarrow C_{2}^{2} = 1 \end{cases}

\Rightarrow 6 + 3 + 1 = 10

165 - 10 = 155

Commented by peter frank last updated on 02/Jul/22

thanks