Your solution is very simple given below if it is not your solution then ask again
In ASPX
<%@ Page Title="Home Page" Language="vb" MasterPageFile="~/Site.Master" AutoEventWireup="false"
CodeBehind="Default.aspx.vb" Inherits="WebApplication4VB._Default" %>
<asp:content id="HeaderContent" runat="server" contentplaceholderid="HeadContent" xmlns:asp="#unknown">
</asp:content>
<asp:content id="BodyContent" runat="server" contentplaceholderid="MainContent" xmlns:asp="#unknown">
<asp:dropdownlist id="dropdown1" runat="server" autopostback="true" onselectedindexchanged="dropdown1_SelectedIndexChanged">
<asp:listitem text="CSC" value="CSC"></asp:listitem>
<asp:listitem text="ECE" value="ECE"></asp:listitem>
<asp:listitem text="EEE" value="EEE"></asp:listitem>
<asp:listitem text="MECH" value="MECH"></asp:listitem>
<asp:listitem text="Other" value="Other"></asp:listitem>
</asp:dropdownlist>
Name of the Department: <asp:textbox id="txtDept" runat="server"></asp:textbox>
</asp:content>
In Code behind .vb
Public Class _Default
Inherits System.Web.UI.Page
Protected Sub Page_Load(ByVal sender As Object, ByVal e As System.EventArgs) Handles Me.Load
End Sub
Protected Sub dropdown1_SelectedIndexChanged(ByVal sender As Object, ByVal e As System.EventArgs) Handles dropdown1.SelectedIndexChanged
txtDept.Text = dropdown1.SelectedValue.ToString()
End Sub
End Class