How-many-5-digits-positive-integers-x-1-x-2-x-3-x-4-x-5-are-there-such-that-x-1-x-2-x-3-x-4-x-5-

Question Number 117061 by ZiYangLee last updated on 09/Oct/20

How many 5 - digits positive integers

x_{1} x_{2} x_{3} x_{4} x_{5} are there such that

x_{1} ⩽ x_{2} ⩽ x_{3} ⩽ x_{4} ⩽ x_{5} ?

Commented by ZiYangLee last updated on 09/Oct/20

n o p e i s 715 i n s t e a d

Commented by MJS_new last updated on 09/Oct/20

2 digits n = \underset{a = 1}{\sum^{9}} a = 45

3 digits n = \underset{b = 1}{\sum^{9}} (\underset{a = 1}{\sum^{b}} a) = 165

4 digits n = \underset{c = 1}{\sum^{9}} (\underset{b = 1}{\sum^{c}} (\underset{a = 1}{\sum^{b}} a)) = 495

5 digits n = \underset{d = 1}{\sum^{9}} (\underset{c = 1}{\sum^{d}} (\underset{b = 1}{\sum^{c}} (\underset{a = 1}{\sum^{b}} a))) = 1287

k digits n = \frac{1}{k!} \underset{j = 0}{\prod^{k - 1}} (j + 9) = \frac{(k + 8)!}{8! k!}

Commented by MJS_new last updated on 09/Oct/20

how you get 715 ?

Answered by Olaf last updated on 09/Oct/20

A number is a plaindrome in a given

base b (often 10 or 16) if its digits are

in nondecreasing order in that base .

A plaindrome in which the digits are

strictly increasing is called metadrome,

while numbers whose digits are

nonincreasing and strictly decreasing

are called nialpdromes and katadromes .

See OEIS, Sequence A 009994 (base 10)

See OEIS, Sequence A 023757 (base 16)

See numbersaplenty . com for calculous .

(in english)

See villemin . gerard . free . fr

(but this is in french)

Answered by mr W last updated on 11/Oct/20

w e s e e i n s u c h a 5 d i g i t n u m b e r t h e

d i g i t z e r o i s n o t a l l o w e d . s o w e c a n

o n l y s e l e c t f r o m d i g i t s 1 t o 9 .

s a y t h e n u m b e r o f d i g i t 1 i s n_{1}, t h e

n u m b e r o f d i g i t 2 i s n_{2}, e t c .

n_{1} + n_{2} + n_{3} + \dots + n_{9} = 5 \dots (i)

w i t h 0 ⩽ n_{1}, n_{2}, \dots, n_{9} ⩽ 5

w e k n o w t h a t w e c a n a l w a y s a r r a n g e

t h e n_{1} t i m e s 1, n_{2} t i m e s 2, \dots, n_{9} t i m e s

9 i n i n c r e a s i n g o r d e r t o f o r m a 5

d i g i t n u m b e r x_{1} x_{2} \dots x_{5} s u c h t h a t

x_{1} ⩽ x_{2} ⩽ \dots ⩽ x_{5} .

s o t h e n u m b e r o f n o n - n e g a t i v e

i n t e g e r s o l u t i o n s o f e q n . (i) i s t h e

a n s w e r o f o u r q u e s t i o n .

t h e n u m b e r o f n o n - n e g a t i v e

i n t e g e r s o l u t i o n s o f e q n . (i) i s t h e

c o e f f i c i e n t o f t e r m x^{5} o f t h e f o l l o w i n g

g e n e r a t i n g f u n c t i o n

{(1 + x + x^{2} + x^{3} + \dots)}^{9} = \frac{1}{{(1 - x)}^{9}} = \underset{k = 0}{\sum^{\infty}} C_{8}^{k + 8} x^{k}

t h e c o f . o f x^{5} t e r m i s C_{8}^{13} = 1287 .

g e n e r a l l y t h e n u m b e r o f k d i g i t

n u m b e r s i s C_{8}^{k + 8} = \frac{(k + 8)!}{8! k!} .

Commented by mr W last updated on 11/Oct/20

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