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Question Number 57042 by mr W last updated on 29/Mar/19

Commented by mr W last updated on 29/Mar/19

$r e p o s t e d$ $Q u e s t i o n f r o m T i n k u t a r a s i r$ $i f o u n d t h e s e u n s o l v e d q u e s t i o n s i n$ $e a r l y p o s t e s a n d r e p o s t t h e m .$
	Answered by mr W last updated on 31/Mar/19
	$(i) number with strictly increasing digits is e.g. 13478. when five digits from {1,2,...,9} are selected, they can always form a number with 5 strictly increasing digits. therefore the answer is C_5 ^9 =126 or we can selectet at most one from each digit from 1 to 9. the answer is the coef. of term x^5 from (1+x)^9 , which is C_5 ^9 . (iii) number with increasing digits is e.g. 12236 or 12222. that means we can select at most 4 from each digit from 1 to 9. the answer is the coef. of term x^5 from (1+x+x^2 +x^3 +x^4 )^9 . (1+x+x^2 +x^3 +x^4 )^9 =(((1−x^5 )^9 )/((1−x)^9 )) =Σ_(k=0) ^9 C_k ^9 (−1)^k x^(5k) Σ_(k=0) ^∞ C_k ^(8+k) x^k term of x^5 is: C_1 ^9 (−1)^1 x^5 C_0 ^8 x^0 +C_0 ^9 (−1)^0 x^0 C_5 ^(13) x^5 =(−C_1 ^9 C_0 ^8 +C_0 ^9 C_5 ^(13) )x^5 =(−C_1 ^9 +C_5 ^(13) )x^5 =(−9+1287)x^5 =1278x^5 ⇒answer is 1278.$
	$(i)$ $n u m b e r w i t h s t r i c t l y i n c r e a s i n g d i g i t s$ $i s e . g . 13478 .$ $w h e n f i v e d i g i t s f r o m {1, 2, . . ., 9} a r e$ $s e l e c t e d, t h e y c a n a l w a y s f o r m a$ $n u m b e r w i t h 5 s t r i c t l y i n c r e a s i n g$ $d i g i t s .$ $t h e r e f o r e t h e a n s w e r i s$ $C_{5}^{9} = 126$ $o r w e c a n s e l e c t e t a t m o s t o n e f r o m$ $e a c h d i g i t f r o m 1 t o 9 .$ $t h e a n s w e r i s t h e c o e f . o f t e r m x^{5}$ $f r o m {(1 + x)}^{9}, w h i c h i s C_{5}^{9} .$ $(i i i)$ $n u m b e r w i t h i n c r e a s i n g d i g i t s i s e . g .$ $12236 o r 12222 .$ $t h a t m e a n s w e c a n s e l e c t a t m o s t 4$ $f r o m e a c h d i g i t f r o m 1 t o 9 .$ $t h e a n s w e r i s t h e c o e f . o f t e r m x^{5}$ $f r o m {(1 + x + x^{2} + x^{3} + x^{4})}^{9} .$ ${(1 + x + x^{2} + x^{3} + x^{4})}^{9} = \frac{{(1 - x^{5})}^{9}}{{(1 - x)}^{9}}$ $= \underset{k = 0}{\sum^{9}} C_{k}^{9} {(- 1)}^{k} x^{5 k} \underset{k = 0}{\sum^{\infty}} C_{k}^{8 + k} x^{k}$ $t e r m o f x^{5} i s :$ $C_{1}^{9} {(- 1)}^{1} x^{5} C_{0}^{8} x^{0} + C_{0}^{9} {(- 1)}^{0} x^{0} C_{5}^{13} x^{5}$ $= (- C_{1}^{9} C_{0}^{8} + C_{0}^{9} C_{5}^{13}) x^{5}$ $= (- C_{1}^{9} + C_{5}^{13}) x^{5}$ $= (- 9 + 1287) x^{5}$ $= 1278 x^{5}$ $\Rightarrow a n s w e r i s 1278 .$
	Commented by mr W last updated on 31/Mar/19

	$(i i)$ $n u m b e r w i t h s t r i c t l y d e c r e a s i n g d i g i t s$ $i s e . g . 76320 .$ $t h a t m e a n s w e c a n s e l e c t 5 d i g i t s f r o m$ $0 t o 9 .$ $n u m b e r o f n u m b e r s w i t h s t r i c t l y$ $d e c r e a s i n g d i g i t s i s$ $C_{5}^{10} = 252 .$ $s t r i c t l y i n c r e a s i n g o r s t r i c t l y d e c r e a s i n g :$ $C_{5}^{9} + C_{5}^{10} = 126 + 252 = 378 .$
	Commented by mr W last updated on 31/Mar/19

	$(i v)$ $n u m b e r w i t h d e c r e a s i n g d i g i t s i s e . g .$ $95331 o r 20000 .$ $t h a t m e a n s w e c a n s e l e c t a t m o s t 4$ $f r o m e a c h d i g i t f r o m 0 t o 9 .$ ${(1 + x + x^{2} + x^{3} + x^{4})}^{10} = \frac{{(1 - x^{5})}^{10}}{{(1 - x)}^{10}}$ $= \underset{k = 0}{\sum^{10}} C_{k}^{10} {(- 1)}^{k} x^{5 k} \underset{k = 0}{\sum^{\infty}} C_{k}^{9 + k} x^{k}$ $t e r m o f x^{5} i s :$ $C_{1}^{10} {(- 1)}^{1} x^{5} C_{0}^{9} x^{0} + C_{0}^{10} {(- 1)}^{0} x^{0} C_{5}^{14} x^{5}$ $= (- C_{1}^{10} C_{0}^{9} + C_{0}^{10} C_{5}^{14}) x^{5}$ $= (- C_{1}^{10} + C_{5}^{14}) x^{5}$ $= (- 10 + 2002) x^{5}$ $= 1992 x^{5}$ $n u m b e r o f n u m b e r s w i t h i n c r e a s i n g$ $a n d d e c r e a s i n g d i g i t s :$ $1278 + 1992 = 3270$
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