<PRE><B>Question 1116837</B>
Assuming the cookies of each type are indistinguishable…. &lt;br&gt;


Counting directly the number of possible ways of selecting 4 cookies with at least one or more chocolate chip cookies:  
 1 choc chip + 3 oatmeal raisin
 2 choc chips + 2 oatmeal raisin
 3 choc chips + 1 oatmeal raisin
  4 choc chips + 0 oatmeal raisin
 
{{{ highlight( 4) }}} ways


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Using stars &amp; bars….&lt;br&gt;

Again assuming indistinguishable cookies within each type,  this problem maps to a stars and bars problem (with one adjustment) where there are two types (k=2) and we must select 4 of them (n=4). &lt;br&gt; 

For k bars (types) and n stars (number of cookies), the arrangements are:
C(n+k-1, k-1) = C(n+k-1, n) &lt;br&gt;

Plugging in n=4, k=2:   C(4+2-1, 1) =  C(5,1) = 5 &lt;br&gt;

The adjustment…
The above answer includes the case no chocolate chip cookies are selected, so we must subtract 1  for that case since it is a condition of the problem that at least one of that type is selected.  That leaves us with 5-1 or {{{ highlight( 4 ) }}} ways.
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It may be tempting to use C(15,4)+C(15,3)+C(15,2)+C(15,1) or other counting formulas but those don&#39;t apply if the cookies of each type are indistinguishable.

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