<PRE><B>Question 1203475</B>
&lt;br&gt;
We will not, on this forum, do your complete assignment for you; the rules of the forum say one problem per post.  And I doubt that we have the capability of providing our answers on the link you provided.&lt;br&gt;
I will explain the first problem for you.  If you want help with the others, you need to post them individually -- entering each question using your keyboard instead of giving a link.&lt;br&gt;
First problem then....&lt;br&gt;
To understand binomial expansion, view the expression {{{(p+q)^n}}} as the product of n factors of {{{p+q}}}.&lt;br&gt;
In multiplying n factors of {{{(p+q)}}}, each partial product is obtained by choosing either the p or the q from each of the n factors.  That means that in each partial product the sum of the exponents on p and q has to be n.&lt;br&gt;
To get a partial product that includes the expression p^n, you need to pick the p from each of the n factors.&lt;br&gt;
To get a partial product that includes p^5, you need to pick the p from 5 of the n factors and the q from the other (n-5) factors.&lt;br&gt;
Consider then a table of the term number, the exponents on p and q, and the coefficient of the term in the full expansion of {{{(p+q)^n}}}.&lt;br&gt;&lt;pre&gt;
  term #  exponent on p   exponent on q  coefficient
 ----------------------------------------------------
    1          n               0          C(n,n) = C(n,0)    [you need to choose &quot;p&quot; in all n factors and &quot;q&quot; in none of them]
    2         n-1              1          C(n,n-1) = C(n,1)  [you need to choose &quot;p&quot; in (n-1) of the n factors and &quot;q&quot; in 1 of them]
    3         n-2              2          C(n,n-2) = C(n,2)  [you need to choose &quot;p&quot; in (n-2) of the n factors and &quot;q&quot; in 2 of them]
   ...
    n          1              n-1         C(n,1)  [you need to choose &quot;p&quot; in 1 of the n factors and &quot;q&quot; in (n-1) of them]
   n+1         0               n          C(n,0)  [you need to choose &quot;p&quot; in 0 of the n factors and &quot;q&quot; in all n of them]&lt;/pre&gt;
Looking at the patterns in the table, we see that the k-th term contains expressions of&lt;br&gt;
(1) {{{q^(k-1)}}}
(2) {{{p^(n-(k-1))}}} ={{{p^(n-k+1)}}}
(3) {{{C(n,k-1)}}}&lt;br&gt;
From that pattern we can see that the first term is p^n and the last term is q^n, so the first of the possible answer choices is true and the next three are not.&lt;br&gt;
And from the pattern we can see that for the k-th = 4th term, k-1 is 4-1 = 3, so the 5th possible answer choice is true and the last one is not.&lt;br&gt;

</PRE>