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 digits


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

$How many 5 - digit numbers from the$ $digits {0, 1, . . . . ., 9} have ?$ $(i) Strictly increasing digits$ $(ii) Strictly increasing or decreasing$ $digits$ $(iii) Increasing digits$ $(iv) Increasing or decreasing digits$
	Answered by aleks041103 last updated on 07/Jun/22
	$let: • the first digit be a • the second − a+b • the 3rd − a+b+c • the 4th − a+b+c+d • the 5th − a+b+c+d+e (i) ⇔ a+b+c+d+e≤9, a,...,e∈{1,...,9} let x_1 =a−1,...,x_5 =e−1 ⇒Σ_(i=1) ^5 x_i ≤4 where x_i ∈{0,...,8} let x_6 =4−Σ_(i=1) ^5 x_i ∈{0,...,4} ⇒we′ve reduced the problem to Σ_(i=1) ^6 x_i =4 this can be done in: C_4 ^(6+4−1) =C_4 ^9 =((9.8.7.6)/(1.2.3.4))=9.7.2=126 (ii) ... (iv) can ve done by analogy$
	$l e t :$ $∙ t h e f i r s t d i g i t b e a$ $∙ t h e s e c o n d - a + b$ $∙ t h e 3 r d - a + b + c$ $∙ t h e 4 t h - a + b + c + d$ $∙ t h e 5 t h - a + b + c + d + e$ $(i) \Leftrightarrow a + b + c + d + e ⩽ 9, a, . . ., e \in {1, . . ., 9}$ $l e t x_{1} = a - 1, . . ., x_{5} = e - 1$ $\Rightarrow \underset{i = 1}{\sum^{5}} x_{i} ⩽ 4$ $w h e r e x_{i} \in {0, . . ., 8}$ $l e t x_{6} = 4 - \underset{i = 1}{\sum^{5}} x_{i} \in {0, . . ., 4}$ $\Rightarrow {w e}^{'} v e r e d u c e d t h e p r o b l e m t o$ $\underset{i = 1}{\sum^{6}} x_{i} = 4$ $t h i s c a n b e d o n e i n : C_{4}^{6 + 4 - 1} = C_{4}^{9} = \frac{9.8.7.6}{1.2.3.4} = 9.7.2 = 126$ $(i i) . . . (i v) c a n v e d o n e b y a n a l o g y$
	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

	$(a) Strictly increasing digits$ $n u m b e r s l i k e 12568 . e a c h d i g i t m a y$ $o c c u r o n l y o n e t i m e . 0 c a n n o t b e$ $u s e d . s o w e c a n o n l y s e l e c t 5 f r o m$ $9 d i g i t s .$ $C_{5}^{9} = 126 .$ $(b) Strictly decreasing digits$ $n u m b e r s l i k e 96540 . e a c h d i g i t m a y$ $o c c u r o n l y o n e t i m e . w e c a n s e l e c t$ $5 f r o m 10 d i g i t s .$ $C_{5}^{10} = 252 .$ $(c) Increasing digits$ $n u m b e r s l i k e 12248 . e a c h d i g i t m a y$ $o c c u r m o r e t i m e s . 0 c a n n o t b e u s e d .$ ${(1 + x + x^{2} + . . .)}^{9} = \frac{1}{{(1 - x)}^{9}} = \underset{k = 0}{\sum^{\infty}} C_{8}^{k + 8} x^{k}$ $c o e f . o f x^{5} i s$ $C_{8}^{5 + 8} = 1287 .$ $(d) D ecreasing digits$ $n u m b e r s l i k e 85521 . e a c h d i g i t m a y$ $o c c u r m o r e t i m e s .$ ${(1 + x + x^{2} + . . .)}^{10} = \frac{1}{{(1 - x)}^{10}} = \underset{k = 0}{\sum^{\infty}} C_{9}^{k + 9} x^{k}$ $c o e f . o f x^{5} i s$ $C_{9}^{5 + 9} = 2002 .$ $s i n c e 00000 i s n o t a v a l i d n u m b e r,$ $s o t h e a n s w e r i s 2002 - 1 = 2001 .$
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