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  1. In how many ways can a given positive integer $n \geq 2$ be expressed as the sum of $2$ positive integers (which are not necessarily distinct). For example, for $n=3$, the number of ways is $2$, i.e., $1+2, 2+1$. Give only the answer without any explanation.
  2. In how many ways can a given positive integer $n \geq 3$ be expressed as the sum of $3$ positive integers (which are not necessarily distinct). For example, for $n=4$, the number of ways is $3$, i.e., $1+2+1, 2+1+1$ and $1+1+2$. Give only the answer without explanation.
  3. In how many ways can a given positive integer $n \geq k$ be expressed as the sum of $k$ positive integers (which are not necessarily distinct). Give only the answer without explanation.
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1 votes

For Distinct  k positive integers : (n+k-1)C(k-1)

For Non-Distinct  k positive integers : (n-1)C(k-1)

1 votes
1 votes

Here, i solved for the generalized case i.e. part c of the question.

 

 

 

 

 

 

 

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0 votes
Here what we have to do is

a) Let the 2 numbers is x1 and x2

    Now it is given that $x1+x2 \ge n$

    x1 and x2 can be anything greater than 0 such that $x1+x2 \ge n$

    minimum value of x1 and x2 is 1 so 1 is a value which is mandatory for x1 and x2

    so, $x1+x2 \ge n $ can be expressed as 10000.... no. of 0s in n-2 times 1 acts as a partition between x1 and x2 . so, no. of   

    ways =   $^{n-2+1}C_1 = n-1$

b) $x1+x2+x3\ge n$ for this it will be $^{n-3+2}C_{2} \implies ^{n-1}C_{2}$

c) similarly for k variables it will be $^{n-1}C_{k-1}$
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0 votes

The above three questions are based on a single logic which is Indistinguishable Objects and Distinguishable Boxes (IODB) why??

Here positive integers don’t matter the same as objects but in which box you are putting that matters for example here there are two boxes in question (a) so both are distinguishable as 1 + 2 and 2 + 1 are different things.

To solve IODB problems think of it as a star-bar problem means objects are stars and the divider between two objects is a bar like in 1 + 2 → 1 and 2 are objects while + is considered as a bar.

Formula used in this IODB template is  $\binom{total number of stars  + total number of bars}{total number of stars}$ = $\binom{total number of stars  + total number of bars}{total number of bars}$

Question a). There are n objects such that value of n >= 2 so at the beginning we will give 2 objects so now we will have n -2 objects or n -2 starts; since we want it as the sum of two positive integers so number of bar = 1.

total number of ways here will be $\binom{n-2 +1}{1}$ { It is simple to use as it is one of the templates in Combinatorics}  

which is equal to  $\binom{n-1}{1}$.

Question b). here there are n –3 stars and two bars so the answer will be  $\binom{n-3 +2}{2}$ =  $\binom{n-1}{2}$.

Question c). here we are already providing k objects so there are left with n – k objects so there are n – k stars and k bars

$\binom{n-k+k -1}{k -1}$ = $\binom{n – 1}{k – 1}$

 { PS:  TO understand IODB template watch Goclasses combinatorics lecture I am not advertising that but that is really awesome. }

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