How-many-5-digit-numbers-from-the-digits-0-1-9-have-i-Strictly-increasing-digits-ii-Strictly-increasing-or-decreasing-digits-iii-Increasing-digits-iv-Increasing-or-decreasing-digit

Question Number 171011 by mr W last updated on 06/Jun/22

How many 5 - digit numbers from the

digits {0, 1, \dots . ., 9} have ?

(i) Strictly increasing digits

(ii) Strictly increasing or decreasing

digits

(iii) Increasing digits

(iv) Increasing or decreasing digits

Answered by aleks041103 last updated on 07/Jun/22

l e t :

∙ t h e f i r s t d i g i t b e a

∙ t h e s e c o n d - a + b

∙ t h e 3 r d - a + b + c

∙ t h e 4 t h - a + b + c + d

∙ t h e 5 t h - a + b + c + d + e

(i) \Leftrightarrow a + b + c + d + e ⩽ 9, a, \dots, e \in {1, \dots, 9}

l e t x_{1} = a - 1, \dots, x_{5} = e - 1

\Rightarrow \underset{i = 1}{\sum^{5}} x_{i} ⩽ 4

w h e r e x_{i} \in {0, \dots, 8}

l e t x_{6} = 4 - \underset{i = 1}{\sum^{5}} x_{i} \in {0, \dots, 4}

\Rightarrow {w e}^{'} v e r e d u c e d t h e p r o b l e m t o

\underset{i = 1}{\sum^{6}} x_{i} = 4

t h i s c a n b e d o n e i n : C_{4}^{6 + 4 - 1} = C_{4}^{9} = \frac{9.8.7.6}{1.2.3.4} = 9.7.2 = 126

(i i) \dots (i v) c a n v e d o n e b y a n a l o g y

Commented by mr W last updated on 08/Jun/22

t h a n k s s i r!

Answered by mr W last updated on 09/Jun/22

(a) Strictly increasing digits

n u m b e r s l i k e 12568 . e a c h d i g i t m a y

o c c u r o n l y o n e t i m e . 0 c a n n o t b e

u s e d . s o w e c a n o n l y s e l e c t 5 f r o m

9 d i g i t s .

C_{5}^{9} = 126 .

(b) Strictly decreasing digits

n u m b e r s l i k e 96540 . e a c h d i g i t m a y

o c c u r o n l y o n e t i m e . w e c a n s e l e c t

5 f r o m 10 d i g i t s .

C_{5}^{10} = 252 .

(c) Increasing digits

n u m b e r s l i k e 12248 . e a c h d i g i t m a y

o c c u r m o r e t i m e s . 0 c a n n o t b e u s e d .

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

c o e f . o f x^{5} i s

C_{8}^{5 + 8} = 1287 .

(d) D ecreasing digits

n u m b e r s l i k e 85521 . e a c h d i g i t m a y

o c c u r m o r e t i m e s .

{(1 + x + x^{2} + \dots)}^{10} = \frac{1}{{(1 - x)}^{10}} = \underset{k = 0}{\sum^{\infty}} C_{9}^{k + 9} x^{k}

c o e f . o f x^{5} i s

C_{9}^{5 + 9} = 2002 .

s i n c e 00000 i s n o t a v a l i d n u m b e r,

s o t h e a n s w e r i s 2002 - 1 = 2001 .

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