Hello, m777!
1(a) Use the Euclidean Algorithm to find $\displaystyle \text{gcd}(190,34)$
I was taught this way . . .
[1] Divide the larger by the smaller; note the remainder.
[2] Divide the remainder into the divisor; note the remainder.
[3] Repeat step 2 until the remainder is 0.
[4] The $\displaystyle \text{gcd}$ is the last divisor. Code:
5 1 1 2 3
------ ---- ---- ---- ---
34 ) 190 20 ) 34 14 ) 20 6 ) 14 2 ) 6
170 20 14 12 ↑ 6
--- --- --- --- --
20 14 6 2 0
Therefore: .$\displaystyle \text{gcd}(190,34) \:=\:2$
1(b) Find $\displaystyle n$ if: $\displaystyle P(n,4) \:=\:42\!\cdot\!P(n,2)$
We have: .$\displaystyle \frac{n!}{(n-4)!} \:=\;42\!\cdot\!\frac{n!}{(n-2)!} \quad\Rightarrow\quad \frac{1}{(n-4)!} \:=$ $\displaystyle \:\frac{42}{(n-2)!}\quad\Rightarrow\quad (n-2)! \:=\:42(n-4)!$
Divide by $\displaystyle (n-4)!\!:\;\;(n-2)(n-3) \:=\:42\quad\Rightarrow\quad n^2 - 5n - 36 \:= \:0$
The quadratic factors: .$\displaystyle (n + 4)(n - 9) \:=\:0$
. . and has two solutions: .$\displaystyle n \:=\:-4,\:9$
Since $\displaystyle n$ must be positive, the answer is: .$\displaystyle \boxed{n \,= \,9}$
2. A computer access code consists of from one to three letters
. . of the English alphabet with repetition allowed.
How many different code words are possible?
Since each code letter has 26 choices,
. . there are: .$\displaystyle 26 \times 26 \times 26 \:=\:17,576$ possible code words.
3(a) A box contains 10 different colored light bulbs.
Find the number of ordered samples of size 3 with replacement.
The first can be any of the 10 bulbs.
The second can be any of the 10 bulbs.
The third can be any of the 10 bulbs.
There are: .$\displaystyle 10 \times 10 \times 10 \:=\:1,\!000$ possible ordered samples.
3(b) How many possible outcomes are there when a fair coin is tossed three times?
The first toss has 2 possible outcomes (Heads or Tails).
The second toss has 2 possible outcomes.
The third toss has 2 possible outcomes.
There are: .$\displaystyle 2 \times 2 \times 2 \:=\:8$ possible outcomes.
4. How many 2-permutations are there of $\displaystyle \{w,x,y,z\}$? .Write them all.
There are: .$\displaystyle P(4,2) \:=\:12$ permutations.
They are: .$\displaystyle \begin{Bmatrix}wx & wy & wz \\ xw & xy & xz \\ yw & yx & yz \\ zw & zx & zy\end{Bmatrix}$