Nucleophilicity refers to the tendency of a species to donate electrons and participate in a nucleophilic reaction. It is influenced by various factors such as the charge, size, electronegativity, and polarizability of the nucleophile. In this question, we need to arrange the given nucleophiles in increasing order of nucleophilicity and explain the reasons behind the order.

Here are the nucleophiles given in the question:
1. Hydroxide ion (OH-)
2. Cyanide ion (CN-)
3. Ethoxide ion (C2H5O-)
4. Methoxide ion (CH3O-)
5. Acetate ion (CH3COO-)

To determine the increasing order of nucleophilicity, we need to consider the following factors:

1. Charge: A nucleophile with a higher negative charge tends to be more nucleophilic because it has more electrons available for donation. Therefore, nucleophiles with a higher negative charge will have higher nucleophilicity.

2. Size: Larger nucleophiles have a greater electron cloud, making it easier for them to donate electrons. Thus, larger nucleophiles tend to be more nucleophilic.

3. Electronegativity: A nucleophile with a lower electronegativity tends to be more nucleophilic because it has less affinity for electrons and is more willing to donate them.

4. Polarizability: Nucleophiles that are more polarizable can easily distort their electron cloud to participate in nucleophilic reactions. Hence, nucleophiles with higher polarizability are generally more nucleophilic.

Now, let's analyze the given nucleophiles based on these factors and arrange them in increasing order of nucleophilicity:

1. Acetate ion (CH3COO-):
- Charge: -1
- Size: Small
- Electronegativity: High (due to the presence of oxygen)
- Polarizability: Low
The acetate ion has a negative charge and is relatively small. However, its high electronegativity and low polarizability make it the least nucleophilic among the given nucleophiles.

2. Methoxide ion (CH3O-):
- Charge: -1
- Size: Small
- Electronegativity: Low (carbon is less electronegative than oxygen)
- Polarizability: High
The methoxide ion has a negative charge and is small in size. Its low electronegativity and high polarizability make it more nucleophilic than the acetate ion.

3. Ethoxide ion (C2H5O-):
- Charge: -1
- Size: Larger than methoxide ion
- Electronegativity: Low
- Polarizability: High
The ethoxide ion has a negative charge and is larger in size compared to the methoxide ion. Its low electronegativity and high polarizability make it more nucleophilic than the methoxide ion.

4. Cyanide ion (CN-):
- Charge: -1
- Size: Larger than ethoxide ion
- Electronegativity: Low
- Polarizability: High
The cyanide ion has a negative charge and is larger than the ethoxide ion. Its low electronegativity and high polarizability make it more nucleophilic than the ethoxide ion.

5. Hydroxide ion (OH-):