Edit in JSFiddle

var directionDisplay;
var directionsService = new google.maps.DirectionsService();
var map;
function initialize()
{
    directionsDisplay = new google.maps.DirectionsRenderer();
    var chicago = new google.maps.LatLng(23.1212, 79.7689);
    var myOptions =
    {
        zoom:5,
        mapTypeId: google.maps.MapTypeId.ROADMAP,
        center: chicago
    }
    map = new google.maps.Map(document.getElementById("map_canvas"), myOptions);
    directionsDisplay.setMap(map);
}
function calcRoute()
{
    var start = document.getElementById("start").value;
    var end = document.getElementById("end").value;
    var request =
    {
        origin:start,
        destination:end,
        provideRouteAlternatives: true,
        travelMode: google.maps.DirectionsTravelMode.DRIVING
    };
    directionsService.route(request, function(response, status)
    {
        if (status == google.maps.DirectionsStatus.OK)
        {
            directionsDisplay.setDirections(response);
          }
    });
}
<html>
<head>
<title>Google Maps API v3 Example: Connecting Indian States</title>
<script type="text/javascript" src="http://maps.google.com/maps/api/js?sensor=false"></script>
<script type="text/javascript">
    <!-- Copy the Javascript code here to try out the code. -->
</script>
</head>
<body onload="initialize()">
<div align="center" style="height: 5%">
<b>Origin: </b>
<select id="start" onchange="calcRoute();">
  <option value="Andhra Pradesh">Andhra Pradesh</option>
  <option value="Arunachal Pradesh">Arunachal Pradesh</option>
  <option value="Assam">Assam</option>
  <option value="Bihar">Bihar</option>
  <option value="Chhattisgarh">Chhattisgarh</option>
  <option value="Goa">Goa</option>
  <option value="Gujarat">Gujarat</option>
  <option value="Haryana">Haryana</option>
  <option value="Himachal Pradesh">Himachal Pradesh</option>
  <option value="Jammu and Kashmir">Jammu and Kashmir</option>
  <option value="Jharkhand">Jharkhand</option>
  <option value="Karnataka">Karnataka</option>
  <option value="Kerala">Kerala</option>
  <option value="Madhya Pradesh">Madhya Pradesh</option>
  <option value="Maharashtra">Maharashtra</option>
  <option value="Manipur">Manipur</option>
  <option value="Meghalaya">Meghalaya</option>
  <option value="Mizoram">Mizoram</option>
  <option value="Nagaland">Nagaland</option>
  <option value="Odisha">Odisha</option>
  <option value="Punjab">Punjab</option>
  <option value="Rajasthan">Rajasthan</option>
  <option value="Sikkim">Sikkim</option>
  <option value="Tamil Nadu">Tamil Nadu</option>
  <option value="Tripura">Tripura</option>
  <option value="Uttar Pradesh">Uttar Pradesh</option>
  <option value="Uttarakhand">Uttarakhand</option>
  <option value="West Bengal">West Bengal</option>
  <option value="Andaman and Nicobar Islands">Andaman and Nicobar Islands</option>
  <option value="Chandigarh">Chandigarh</option>
  <option value="Dadra and Nagar Haveli">Dadra and Nagar Haveli</option>
  <option value="Daman and Diu">Daman and Diu</option>
  <option value="Lakshadweep">Lakshadweep</option>
  <option value="Delhi">Delhi</option>
  <option value="Pondicherry">Pondicherry</option>
</select>
<b>Destination: </b>
<select id="end" onchange="calcRoute();">
  <option value="Andhra Pradesh">Andhra Pradesh</option>
  <option value="Arunachal Pradesh">Arunachal Pradesh</option>
  <option value="Assam">Assam</option>
  <option value="Bihar">Bihar</option>
  <option value="Chhattisgarh">Chhattisgarh</option>
  <option value="Goa">Goa</option>
  <option value="Gujarat">Gujarat</option>
  <option value="Haryana">Haryana</option>
  <option value="Himachal Pradesh">Himachal Pradesh</option>
  <option value="Jammu and Kashmir">Jammu and Kashmir</option>
  <option value="Jharkhand">Jharkhand</option>
  <option value="Karnataka">Karnataka</option>
  <option value="Kerala">Kerala</option>
  <option value="Madhya Pradesh">Madhya Pradesh</option>
  <option value="Maharashtra">Maharashtra</option>
  <option value="Manipur">Manipur</option>
  <option value="Meghalaya">Meghalaya</option>
  <option value="Mizoram">Mizoram</option>
  <option value="Nagaland">Nagaland</option>
  <option value="Odisha">Odisha</option>
  <option value="Punjab">Punjab</option>
  <option value="Rajasthan">Rajasthan</option>
  <option value="Sikkim">Sikkim</option>
  <option value="Tamil Nadu">Tamil Nadu</option>
  <option value="Tripura">Tripura</option>
  <option value="Uttar Pradesh">Uttar Pradesh</option>
  <option value="Uttarakhand">Uttarakhand</option>
  <option value="West Bengal">West Bengal</option>
  <option value="Andaman and Nicobar Islands">Andaman and Nicobar Islands</option>
  <option value="Chandigarh">Chandigarh</option>
  <option value="Dadra and Nagar Haveli">Dadra and Nagar Haveli</option>
  <option value="Daman and Diu">Daman and Diu</option>
  <option value="Lakshadweep">Lakshadweep</option>
  <option value="Delhi">Delhi</option>
  <option value="Pondicherry">Pondicherry</option>
</select>
</div>
<div id="map_canvas" style="top:20px; height: 400px"></div>
</body>
</html>