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

FacebookTweetPin

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

$$\mathrm{How}\mathrm{many}5-\mathrm{digit}\mathrm{numbers}\mathrm{from}\mathrm{the}$$$$\mathrm{digits}\{0,1,\dots ..,9\}\mathrm{have}?$$$$\left(\mathrm{i}\right)\mathrm{Strictly}\mathrm{increasing}\mathrm{digits}$$$$\left(\mathrm{ii}\right)\mathrm{Strictly}\mathrm{increasing}\mathrm{or}\mathrm{decreasing}$$$$\mathrm{digits}$$$$\left(\mathrm{iii}\right)\mathrm{Increasing}\mathrm{digits}$$$$\left(\mathrm{iv}\right)\mathrm{Increasing}\mathrm{or}\mathrm{decreasing}\mathrm{digits}$$

Answered by aleks041103 last updated on 07/Jun/22

$$let:$$$$\bullet thefirstdigitbea$$$$\bullet thesecond-a+b$$$$\bullet the3rd-a+b+c$$$$\bullet the4th-a+b+c+d$$$$\bullet the5th-a+b+c+d+e$$$$\left(i\right)\iff a+b+c+d+e⩽9,a,\dots ,e\in \{1,\dots ,9\}$$$$let{x}_1=a-1,\dots ,x_5=e-1$$$$\Rightarrow \underset{i=1}{\sum ^5}{x}_i⩽4$$$$where{x}_i\in \{0,\dots ,8\}$$$$let{x}_6=4-\underset{i=1}{\sum ^5}{x}_i\in \{0,\dots ,4\}$$$$\Rightarrow {we}^{\prime }vereducedtheproblemto$$$$\underset{i=1}{\sum ^6}{x}_i=4$$$$thiscanbedonein:C_4^{6+4-1}=C_4^9=\frac{9.8.7.6}{1.2.3.4}=9.7.2=126$$$$\left(ii\right)\dots \left(iv\right)canvedonebyanalogy$$$$$$

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

$$thankssir!$$

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

$$\left(a\right)\mathrm{Strictly}\mathrm{increasing}\mathrm{digits}$$$$numberslike12568.eachdigitmay$$$$occuronlyonetime.0cannotbe$$$$used.sowecanonlyselect5from$$$$9digits.$$$$C_5^9=126.$$$$$$$$\left(b\right)\mathrm{Strictly}\mathrm{decreasing}\mathrm{digits}$$$$numberslike96540.eachdigitmay$$$$occuronlyonetime.wecanselect$$$$5from10digits.$$$$C_5^{10}=252.$$$$$$$$\left(c\right)\mathrm{Increasing}\mathrm{digits}$$$$numberslike12248.eachdigitmay$$$$occurmoretimes.0cannotbeused.$$$${(1+x+x^2+\dots )}^9=\frac{1}{{(1-x)}^9}=\underset{k=0}{\sum ^{\mathrm{\infty }}}{C}_8^{k+8}{x}^k$$$$coef.of{x}^{5}is$$$$C_8^{5+8}=1287.$$$$$$$$\left(d\right)D\mathrm{ecreasing}\mathrm{digits}$$$$numberslike85521.eachdigitmay$$$$occurmoretimes.$$$${(1+x+x^2+\dots )}^{10}=\frac{1}{{(1-x)}^{10}}=\underset{k=0}{\sum ^{\mathrm{\infty }}}{C}_9^{k+9}{x}^k$$$$coef.of{x}^{5}is$$$$C_9^{5+9}=2002.$$$$since00000isnotavalidnumber,$$$$sotheansweris2002-1=2001.$$

Terms of Service
Privacy Policy
Contact: info@tinkutara.com

FacebookTweetPin