ABCD-is-four-digits-integers-How-many-ABCD-that-suitable-with-A-B-C-D-25-

Question Number 57594 by naka3546 last updated on 08/Apr/19

A B C D i s f o u r d i g i t s i n t e g e r s .

H o w m a n y A B C D t h a t s u i t a b l e w i t h A + B + C + D = 25 ?

Answered by mr W last updated on 08/Apr/19

A : 1, 2, . ., 9

B, C, D : 0, 1, 2, . ., 9

A + B + C + D = 25

(x + x^{2} + x^{3} + \dots + x^{9}) {(1 + x + x^{2} + x^{3} + \dots + x^{9})}^{3}

= \frac{x (1 - x^{9}) {(1 - x^{10})}^{3}}{{(1 - x)}^{4}}

= \frac{x (1 - x^{9}) (1 - 3 x^{10} + 3 x^{20} - x^{30})}{{(1 - x)}^{4}}

= \frac{x (1 - x^{9} - 3 x^{10} + 3 x^{19} + 3 x^{20} + \dots)}{{(1 - x)}^{4}}

= x (1 - x^{9} - 3 x^{10} + 3 x^{19} + 3 x^{20} + \dots) \underset{k = 0}{\sum^{\infty}} C_{3}^{k + 3} x^{k}

c o e f . o f x^{25} i s

C_{3}^{27} - C_{3}^{18} - 3 C_{3}^{17} + 3 C_{3}^{8} + 3 C_{3}^{7} = 2925 - 816 - 3 \times 680 + 3 \times 56 + 3 \times 35 = 342

\Rightarrow t h e r e a r e 342 s u i t a b l e n u m b e r s .

Commented by mr W last updated on 08/Apr/19

f o r m o r e d e t a i l s s e e Q 21800 .

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