Define logic and explain the logical operators conjunction, disjunction, negation, and bi-implication. Prove De Morgan's law for disjunction using a truth table.

For $n=1$:
LHS: 1
RHS: $\frac{1(1+1)}{2} = \frac{1 \times 2}{2} = 1$
Since LHS = RHS, the formula holds for $n=1$.

Assume the formula holds for some positive integer $k$.

We need to show that the formula holds for $n=k+1$. That is, we need to show:

Start with the LHS:

Using the inductive hypothesis, substitute the sum up to $k$:

Find a common denominator:

Factor out $(k+1)$:

This is the RHS.

By the principle of mathematical induction, the formula $1+2+3+\dots+n = \frac{n(n+1)}{2}$ holds for all positive integers $n$.

2b) To expand $(a+b)^6$ using Pascal's triangle, we first find the coefficients from the 6th row of Pascal's triangle.
Row 0: 1
Row 1: 1 1
Row 2: 1 2 1
Row 3: 1 3 3 1
Row 4: 1 4 6 4 1
Row 5: 1 5 10 10 5 1
Row 6: 1 6 15 20 15 6 1

Using these coefficients, the expansion is:

3a)

• Permutation: A permutation is an arrangement of objects in a specific order. The order of selection or arrangement matters. For example, if we select two letters from A, B, C, the permutations are AB, BA, AC, CA, BC, CB.
• Combination: A combination is a selection of objects where the order of selection does not matter. It is simply a group or subset of items. For example, if we select two letters from A, B, C, the combinations are {A, B}, {A, C}, {B, C}.

3b) To find the number of permutations from the letters of the word STATISTICS, we first count the total number of letters and the frequency of each repeated letter.
The word is STATISTICS.
Total number of letters, $n = 10$.

Frequencies of repeated letters:

• S appears 3 times ($n_1 = 3$)
• T appears 3 times ($n_2 = 3$)
• A appears 1 time
• I appears 2 times ($n_3 = 2$)
• C appears 1 time

The formula for permutations with repetitions is $\frac{n!}{n_1! n_2! \dots n_k!}$.

$n = 10$
$n_S = 3$
$n_T = 3$
$n_I = 2$

$10! = 3,628,800$
$3! = 3 \times 2 \times 1 = 6$
$2! = 2 \times 1 = 2$

The number of permutations that can be made from the letters of the word STATISTICS is $\boxed{50,400}$.