How to attach recaptcha to ajax form?

Question

PHP

Essence of the question. There is a feedback form without reloading the page on AJAX. I would like to put there a recaptcha from Google. How to implement it. Preferably with examples, otherwise I'm a noob.

<form class="form-horizontal" target="_blank" method="post" id="form">
            <div class="form-group">
              <label  class="col-sm-2 control-label">Ф.И.О.:</label>
              <div class="col-sm-10">
              <input class="form-control" type="text" name="name" placeholder="Имя"  required />
              </div>
            </div>
            <div class="form-group">
              <label  class="col-sm-2 control-label">Телефон:</label>
              <div class="col-sm-10">
              <input class="form-control" type="text" name="number" pattern="^[0-9]{11}$" placeholder="Телефон"  required />
              </div>
            </div>
              <div class="form-group">
              <label class="col-sm-2 control-label">Эл. почта:</label>
              <div class="col-sm-10">
                <input class="form-control"  type="email" name="email" placeholder="E-mail"  required />
              </div>
            </div>
            <div class="form-group">
              <div class="col-sm-offset-2 col-sm-10">
                <textarea class="form-control" rows="3" name="message" placeholder="Ваше сообщение"  required></textarea>	
              </div>
            </div>
            <div class="form-group">
              <div class="col-sm-offset-2 col-sm-10">
                <button class="btn btn-default">Отправить</button>
                <label class="sucsses">СООБЩЕНИЕ ОТПРАВЛЕННО</label>
              </div>
            </div>
          </form>

$("#form").submit(function() {
        $.ajax({
            type: "POST",
            url: "http://qwerty.developer33.ru/wp-content/themes/Alliance/mail.php",
            data: $(this).serialize()
        }).done(function() {
            $(this).find("input").val("");
            $(".sucsses").fadeIn("1500");
            setTimeout(function() { $(".sucsses").fadeOut("1500"); }, 2500);   
            $("#form").trigger("reset");
        });
        return false;
    });

<?php

$recepient = "[email protected] ";

$name = trim($_POST["name"]);
$number = trim($_POST["number"]);
$email = trim($_POST["email"]);
$message = trim($_POST["message"]);

$message = "Имя: $name \nТелефон: $number \nЕmail: $email \nСообщение: $message";

$headers = 'Content-type: text/plain; charset="utf-8"';
$headers .= "MIME-Version: 1.0\r\n";
$headers .= "From: [email protected]\r\n";
$headers .= "Reply-To: [email protected]\r\n";

$pagetitle = "Сообщение с АЛЬЯНС-сайта";
mail($recepient, $pagetitle, $message, $headers);
?>

code attached

Answer 1

I would like to put there a recaptcha from Google.

Why google recaptcha?
It's not easier to put something like this (you need to click on the "search" button)