total no. in each cases

juantheron · 2011-12-15 04:03:07

Let N be a no. of 4 digit numbers a b c d . where the digit satisfy the condition

(i) a<b<c<d

(ii) a>b>c>d

(iii) a<=b<=c<=d

(iv) a>=b>=c>=d

these are 4 different cases

then find total no. N in each cases

bobbym · 2011-12-15 06:37:31

Hi;

i) This is the number of solutions to a+b+c+d =r with a<b<c<d and 10>a>0, 10>b,c,d>=0

ii) This is the number of solutions to a+b+c+d =r with a>b>c>d and 10>a>0, 10>b,c,d>=0

iii) This is the number of solutions to a+b+c+d =r with a<=b<=c<=d and 10>a>0, 10>b,c,d>=0

iv)This is the only one that has any theory connected with it that I am aware of. The generating function is:

The coefficient of x^4 = 714 so there 714 numbers like that. The formula is:

juantheron · 2011-12-15 15:34:12

thanks bobbym

but how you get these things

bobbym · 2011-12-15 15:38:42

Hi;

i,ii,iii are known to me because I do a lot of work with generating functions. Basically they had to be counted by a computer program. The binomials were surmised by playing spot the pattern.

iv) has a theoretical background. It comes from books. How it is derived exactly I do not know.

juantheron · 2011-12-15 16:02:15

thanks bobbym . is there is any link in which all things are present

bobbym · 2011-12-15 16:12:35

Hi;

I would have used it and I did look. For the fourth one there is a mention of it at the OEIS but that requires that you already have the answer by some other method.

Math Is Fun Forum

#1 2011-12-15 04:03:07

total no. in each cases

#2 2011-12-15 06:37:31

Re: total no. in each cases

#3 2011-12-15 15:34:12

Re: total no. in each cases

#4 2011-12-15 15:38:42

Re: total no. in each cases

#5 2011-12-15 16:02:15

Re: total no. in each cases

#6 2011-12-15 16:12:35

Re: total no. in each cases

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