Combinatorics

We are given the following sums: We need to find the value of .

1. Step 1: Use the binomial expansion: The binomial expansion of is: This simplifies to:

2. Step 2: Split the sum into even and odd terms: The sum of the even-indexed binomial coefficients is: and the sum of the odd-indexed binomial coefficients is: By symmetry of binomial coefficients, we know that:

3. Step 3: Conclusion: Since , it follows that: Thus, the correct answer is , corresponding to option (A).

The set has 6 elements. The total number of subsets of a set of size is . For this set, the total number of subsets is: Now, we need to find the number of subsets with an odd number of elements. By symmetry, half of the subsets will have an odd number of elements, and half will have an even number of elements. Therefore, the number of subsets with an odd number of elements is:
Thus, the correct answer is (B) .

We are tasked with finding how many 3-digit numbers greater than 500 can be made using the digits 3, 4, 5, and 7. Step 1: Consider the first digit.
For the number to be greater than 500, the first digit must be 5 or 7, as these are the only digits from the given set that will make the number greater than 500. So, there are 2 possible choices for the first digit: or .
Step 2: Consider the second and third digits.
For the second and third digits, we can use any of the 4 available digits (3, 4, 5, 7) because there are no restrictions. Thus, for each of the second and third digits, there are 4 possible choices.
Step 3: Calculate the total number of possibilities.
Now, multiply the number of possibilities for each digit: Thus, the total number of 3-digit numbers greater than 500 that can be made with the digits 3, 4, 5, and 7 is:

We are tasked with finding the number of combinations of 3-lettered numbers using the digits 3, 4, 5, and 7, with repetition allowed. Additionally, the digits cannot be equal to zero. The number of possible choices for each digit is 4 (since we are choosing from the set {3, 4, 5, 7}). Step 1: Apply the formula for combinations with repetition.
When repetition is allowed, the number of combinations for selecting items from items is given by the formula: where is the number of available choices for each item, and is the number of items to be selected. Here, (since we have 4 available digits: 3, 4, 5, 7) and (since we are selecting 3 digits for each number).
Step 2: Calculate the total number of combinations.
Using the formula , the total number of combinations is: Thus, the number of 3-lettered numbers that can be created is:

Concept:
Algebra - Binomial Expansion Identity.

Step 1: Identify the components of the binomial expansion.
Here, let , , and .
Step 2: Apply the identity formula.
Since we are adding conjugate binomials to the 5th power, the odd power terms cancel out: $ 1 \cdot x^{10} \cdot 1 = x^{10} 10 \cdot x^6 \cdot (1-x^2) = 10x^6 - 10x^8 5 \cdot x^2 \cdot (1-x^2)^2 = 5x^2(1 - 2x^2 + x^4) = 5x^2 - 10x^4 + 5x^6 x^8 x^8 x^{10} x^8 -10x^8 -10 x^6 x^8 2 $

Concept:
Combinatorics - Permutations.
A key property of permutations linking consecutive values is: .
Step 1: Apply the permutation property to the given equations.
Using the property , substitute : $ ^nP_5 = 6720 ^{(n-1)}P_4 = 840 n = 8 ^nP_3 ^8P_3 $

Concept:
Geometry / Combinatorics - Diagonals of a Polygon.
The number of diagonals in an -sided polygon is given by the formula .
Step 1: Set up the equation using the given formula.
Let be the number of sides of the polygon. $ -340 -3 -20 17 n n = -17 $

Concept:
Combinatorics - Combinations Formula.
The formula for combinations is , which simplifies to descending terms of divided by .
Step 1: Write out the combination formula for .
Expand using the descending factorial method: $ n n(n-1)(n-2)(n-3) 13, 7, 5, 3, 8, 3 13 14 7 \times 2 15 5 \times 3 4 \times 3 = 12 $

Step 1: Understanding the Concept:
The number of subsets with exactly elements from a set of elements is given by the binomial coefficient .
Step 2: Key Formula or Approach:
Required number for .
Step 3: Detailed Explanation:
Total subsets for is .
Sum of all coefficients: .
By symmetry: , , , , .
Thus, is exactly half of the total sum.
Result .
Step 4: Final Answer:
The number of subsets is 256.