# if 1 <=r<=n prove that ncr + ncr-1 = ncr +n+1cr+1

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### 1 Answer

Combinations refers to number of ways to select "k" objects from a "n" collection. The formula for combination of *n* things taken *k* at a time without repetition follows the formula:

`_nC_k` or C(n,k) =` (n!)/(k!(n-k)!)`

With the given condition `1<=r<=n` , it shows that r and n are both non-negative integers greater than 1.` `

It coincides with the definition of factorial of non-negative integer "n" is the product of all positive integers less than or equal to n.

Factorial notation: `n= n*(n-1)!`

For the given, please double check the expression at the right-hand side.

They are not equal.

To clarify this, we may let n=7 and r =5.

Plug-in the values as shown:

`_nC_r + _nC_(r-1) ` =? `_nC_r + _(n+1)C_(r+1)`

`_7C_5 + _(7)C_(5-1)` =?` _7C_5 + _(7+1)C_(5+1)`

`_7C_5 + _7C_4 ` = ?`_7C_5 + _8C_6`

21 + 35 =? 21 + 28

56 ≠ 49

In addition, it can be simplified as:

`_nC_r +_nC_(r-1)` = `_nC_r + _(n+1)C_(r+1)`

- `_nC_r ` - `_nC_r`

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`_nC_(r-1)` = `_(n+1)C_(r+1)`

It will still not be an equal values. To check, we let n=7 and r=5 :

** ` ` = ` `**

`_7C_(5-1)` =?` _(7+1)C_(5+1)`

`_7C_4` =? `_8C_6`

35 =? 28

35 ≠ 28

It might have been given as` _nC_r + _nC_(r-1)` ** =**`_(n+1)C_r` **` `**

Starting at the left-hand side (LHS), we may apply the combination formula as:

`_nC_r + _nC_(r-1)` = `(n!)/(r!(n-r)!) +(n!)/(r!(n-(r-1))!)`

=`(n!)/(r!(n-r)!) +(n!)/((r-1)!(n-r+1)!) ` Simplify: `(n-(r-1)) =(n-r+1)`

=`(n!)/(r*(r-1)!(n-r)!) +(n!)/((r-1)!(n-r+1)(n-r)!)`

Note

:` r! = r*(r-1)!` and

`(n-r+1)! = (n-r+1)(n-r+1-1)! = (n-r+1)(n-r)!`

Factoring out `(n!)/((r-1)!(n-r)!)` :

`(n!)/((r-1)!(n-r)!) * ( 1/r +1/(n-r+1)) = (n!)/((r-1)!(n-r)!) * ( (n-r+1) +r)/(r(n-r+1)) `

=`(n!)/((r-1)!(n-r)!) * (n+1)/(r(n-r+1)) `

= `((n+1)(n!))/(r(r-1)!(n-r+1)(n-r)!) `

= `((n+1)!)/(r!(n-r+1)!) ` or `((n+1)!)/(r!((n+1)-r)!)`

= `_(n+1)C_r`

Proof: LHS = RHS

`_nC_r + _nC_(r-1)` = _(n+1)C_r

`_(n+1)C_r` =`_(n+1)C_r`

Checking using n=7 and r=5 shows:

`_7C_5 + _7C_(5-1)` =? ` _(7+1)C5`

`_7C_5 + _7C_4 ` =? `_8C_5`

21 +35 =? 56

56 = 56

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