The correct match of the following is :
`{:(,"Column I",,"Column II"),((i),|__ul(0),(a),(|__ul(n))/(|__ul(n-r))),((ii),|__ul(n),(b),1),((iii),""^(n)C_(r ),(c ),1xx2xx3xx....xxn),((iv),""^(n)P_(r ),(d),(|__ul(n))/(|__ul(r )|__ul(n-r))):}`

Options

A. (i)-(b), (ii)-(c ), (iii)-(d), (iv)-(a)
B. (i)-(d), (ii)-(a), (iii)-(c ), (iv)-(b)
C. (i)-(c ), (ii)-(b), (iii)-(a), (iv)-(d)
D. (i)-(a), (ii)-(d),(iii)-(c ),(iv)-(b)

Answer

To solve the problem of matching the items in Column I with those in Column II, we will analyze each item in Column I and find its corresponding expression in Column II.

### Step-by-step Solution:

1. **Identify the first item in Column I:**
- The first item is \(0!\) (zero factorial).
- By definition, \(0! = 1\).

2. **Match \(0!\) with Column II:**
- In Column II, we see that the value \(1\) corresponds to option (b).
- Therefore, we can match \(0!\) with (b).

3. **Identify the second item in Column I:**
- The second item is \(n!\) (n factorial).
- By definition, \(n! = n \times (n-1) \times (n-2) \times \ldots \times 3 \times 2 \times 1\).
- This can also be expressed as \(1 \times 2 \times 3 \times \ldots \times n\).

4. **Match \(n!\) with Column II:**
- In Column II, this expression corresponds to option (c).
- Therefore, we can match \(n!\) with (c).

5. **Identify the third item in Column I:**
- The third item is \(\binom{n}{r}\) (n choose r or combinations).
- The formula for combinations is given by \(\binom{n}{r} = \frac{n!}{r!(n-r)!}\).

6. **Match \(\binom{n}{r}\) with Column II:**
- In Column II, this expression corresponds to option (d).
- Therefore, we can match \(\binom{n}{r}\) with (d).

7. **Identify the fourth item in Column I:**
- The fourth item is \(P(n, r)\) (permutations).
- The formula for permutations is given by \(P(n, r) = \frac{n!}{(n-r)!}\).

8. **Match \(P(n, r)\) with Column II:**
- In Column II, this expression corresponds to option (a).
- Therefore, we can match \(P(n, r)\) with (a).

### Final Matches:
- \(0! \rightarrow (b)\)
- \(n! \rightarrow (c)\)
- \(\binom{n}{r} \rightarrow (d)\)
- \(P(n, r) \rightarrow (a)\)

### Summary of Matches:
1. \(0! \rightarrow (b)\)
2. \(n! \rightarrow (c)\)
3. \(\binom{n}{r} \rightarrow (d)\)
4. \(P(n, r) \rightarrow (a)\)