Don't go for code behind, this can be possible in Javascript only.
<script type="text/javascript">
function getDropdownListSelectedText()
{
var DropdownList=document.getElementById('<%=aspdropdown.ClientID %>');
var SelectedIndex=DropdownList.selectedIndex;
var SelectedValue=DropdownList.value;
var SelectedText=DropdownList.options[DropdownList.selectedIndex].text;
var TextBox=document.getElementById('<%=TextBox.ClientID %>');
var sValue='Index: '+SelectedIndex+' Selected Value: '+SelectedValue+' Selected Text: '+SelectedText;
TextBox.innerHTML=sValue;
}
</script>