The determinant of the matrix $$$\left[ {\matrix{ 2 & 0 & 0 & 0 \cr 8 & 1 & 7 & 2 \cr 2 & 0 & 2 & 0 \cr 9 & 0 & 6 & 1 \cr } } \right]\,\,is$$$

Let $$S = \left\{ {0,1,2,3,4,5,6,7} \right\}$$ and $$ \otimes $$ denote multiplication modulo $$8$$, that is, $$x \otimes y = \left( {xy} \right)$$ mod $$8$$

(a) Prove that $$\left( {0,\,1,\, \otimes } \right)$$ is not a group.
(b) Write $$3$$ distinct groups $$\left( {G,\,\, \otimes } \right)$$ where $$G \subset s$$ and $$G$$ has $$2$$ $$\,\,\,\,\,\,$$elements.

A multiset is an unordered collection of elements where elements may repeat ay number of times. The size of a multiset is the number of elements in it counting repetitions.

(a) what is the number of multisets of size 4 that can be constructed from n distinct elements so that at least one element occurs exactly twice?
(b) How many multisets can be constructed from n distinct elements?

$${{E_1}}$$ and $${{E_2}}$$ are events in a probability space satisfying the following constraints:
$$ \bullet $$ $$\Pr \,\,({E_1}) = \Pr \,({E_2})$$
$$ \bullet $$ $$\Pr \,\,({E_1}\, \cup {E_2}) = 1$$
$$ \bullet $$ $${E_1}$$ & $${E_2}$$ are independent

The value of Pr ($${E_1}$$), the probability of the event $${E_1}$$, is