Here is a solution for randomising class names using pure JavaScript.
Updated answer
I have updated my solution now that the question was clarified, here is it adapted to your colors. I have set the background-color of the .cards to the colors set in the array. This could easily be done using the id as well, I recommend against using [] characters in an id though as I think I'm not sure if that's standards compliant.
jsFiddle

var colors = [
"orange","orange","pink","pink","red","red","purple","purple",
"blue","blue","green","green","brown","brown","yellow","yellow"
];
var divs = document.getElementsByClassName("card");
while (divs.length > 0) {
var i = Math.floor(Math.random() * colors.length);
divs[0].style.backgroundColor = colors[i];
colors.splice(i, 1);
divs = [].slice.call(divs, 1);
}
Original answer
Given an array of ids and a set of HTML elements, a random id will be assigned to each element from ids.
jsFiddle

JavaScript
var ids = ["1", "2", "3", "4", "5", "6", "7", "8", "9", "10"];
var divs = document.getElementsByClassName("game-element");
while (divs.length > 0) {
var i = Math.floor(Math.random() * ids.length);
divs[0].id = 'item-' + ids[i];
ids.splice(i, 1);
divs = [].slice.call(divs, 1);
}
HTML
<div class="game-element"></div>
<div class="game-element"></div>
<div class="game-element"></div>
<div class="game-element"></div>
<div class="game-element"></div>
<div class="game-element"></div>
<div class="game-element"></div>
<div class="game-element"></div>
<div class="game-element"></div>
<div class="game-element"></div>
CSS
.game-element {
width:10px;
height:10px;
float:left;
}
#item-1 { background-color:#F00; }
#item-2 { background-color:#0F0; }
#item-3 { background-color:#00F; }
#item-4 { background-color:#FF0; }
#item-5 { background-color:#F0F; }
#item-6 { background-color:#0FF; }
#item-7 { background-color:#A0A; }
#item-8 { background-color:#0AA; }
#item-9 { background-color:#AA0; }
#item-10 { background-color:#000; }