${}^{k}C_{r}+{}^{k}C_{r-1}=$

Question

Accepted Answer

Explanation:

To solve the given problem, we need to understand the properties of combinations. The expression given is:

$$ {}^{k}C_{r} + {}^{k}C_{r-1} $$

This expression can be simplified using the Pascal's Identity, which states:

$$ {}^{n}C_{r} + {}^{n}C_{r-1} = {}^{n+1}C_{r} $$

In this identity, the left side represents the sum of two combinations from the same set of elements, but with different numbers of elements chosen. The right side shows that this sum is equivalent to choosing $ r $ elements from a set of $ n+1 $ elements.

Applying Pascal's Identity to the given expression:

$$ {}^{k}C_{r} + {}^{k}C_{r-1} = {}^{k+1}C_{r} $$

This matches Option 3, which is:

3. ${}^{k+1}C_{r}$

Therefore, Option 3 is correct because it directly follows from Pascal's Identity, which is a fundamental property of combinations.

Now, let's explain why the other options are incorrect:

1. **Option 1: ${}^{k}C_{r+1}$**

This option suggests that the sum of the combinations results in choosing $ r+1 $ elements from the original $ k $ elements, which does not align with Pascal's Identity. The identity specifically results in choosing the same number of elements, $ r $, from an increased set size, $ k+1 $.

2. **Option 2: ${}^{k-1}C_{r+1}$**

This option implies reducing the total number of elements to $ k-1 $ and increasing the number of elements chosen to $ r+1 $. This does not follow from the given expression or Pascal's Identity, which involves increasing the total number of elements by one, not decreasing it.

3. **Option 4: ${}^{k+1}C_{r+1}$**

This option suggests both increasing the total number of elements and the number of elements chosen by one. However, Pascal's Identity only increases the total number of elements, not the number of elements chosen. Therefore, this option does not correctly represent the identity used to simplify the given expression.

In conclusion, Option 3 is the correct answer because it accurately reflects the application of Pascal's Identity to the given combination expression.