self doubt

Question

we use the formula V(n,k) = C(n-1+k,k)

when do we actually go for this formula?

QUESTION

there are 6 symbols and we have to insert 12 blanks between these symbols with atleast 2 blanks between each...number of ways to arrange the blanks

Solution

remaining blanks =2 after keeping 2 in between each

V(5,2) = C(5-1+2,2) =15

why use this approach

in simple words i want to know when do we go for this approach

Comments

This is known as: Bars and stars method https://en.wikipedia.org/wiki/Stars_and_bars_(combinatorics)

This article is good: https://brilliant.org/wiki/integer-equations-star-and-bars/

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you have 6 symbols , and b/w any two at least two space must be there , so first take care about those two and place exactly two space b/w consecutive numbers.b/w 6 numbers there are 5 places and now 2 spaces remain so it's eventually like

x₁+x₂+x₃+x₄+x₅=2

which is (5-1+2)C2

Answer 1

This can be also rephrased as "chocolates in box" problem..Specifically , in this problem , we distribute identical objects to distinct boxes or locations ..

Here it can be applied as the gap between the symbols can be treated as distinct each of which will be treated as a box and the spaces which are to be filled in these gaps (boxes) can be treated as objects.And we will fill using space character only , hence the objects are of identical nature..

So the solution to such problem is the same as finding number of non negative integral solutions of the equation :

X1 + X2 + ..........+ Xn   =   r where 'r' objects and 'n' boxes are there and each box can have any non negative(including 0 number of objects) in it..

Hence the given problem can be written as :

X1 + X2 + ........ + X5    =    12   where each of the variables follow  Xi >= 2 for  1 <= i <= 5..Hence to reduce it to standard form we rewrite it as :

         X1 + 2 + X2 + 2 .... + X5 + 2   =   12    , where     Xi >= 0 for 1 <= i <= 5

==>   X1 + X2 ......  + X5                 =    12  -  2 * 5   =  12 - 10    =   2

==>    X1 + X2 ......  + X5                 =    2

Hence  n =  5  ,  r  =  2..

Hence number of non negative integral solutions of the equation   =   ^n-1+rC_r

                                                                                               =  ^5-1+2C₂

                                                                                               =  ⁶C₂

                                                                                               =  15

Hence number of ways of filling the spaces in between the symbols    =   15