How To Use objectAtIndex In an Array Object

I see a few issues here. The immediate cause of your compiler issue is that you are trying to call NSMutableArray methods on an instance of DeckOfCards. Instead, you want toc call these methods on the NSMutableArray returned by the method-[DeckOfCards deckOfCards]. It would look something like this:

DeckOfCards *newDeck = [[DeckOfCards alloc] init];
NSMutableArray *deckArray = [newDeck deckOfCards];
int r = arc4random() % 52;
NSLog (@"The card you picked is: %@, and there are %i cards left",[deckArray objectAtIndex:r], [deckArray count]);
[deckArray removeObjectAtIndex:r];

But this leads us to our next issue - -deckOfCards doesn't actually return anything - its return type is defined as void. So in order to get that NSMutableArray instance back to the rest of your program you're going to have to return it, and define the method with the appropriate return type. If we were just to address this issue, we'd change your -deckOfCards method like so:

-(NSMutableArray *)deckOfCards
{
    NSMutableArray *deckOfCards = [NSMutableArray arrayWithObjects:
        @"One of Hearts", @"Two of Hearts"..., nil];

    return deckOfCards;
}

OK, so that should compile now. But there are some serious errors here. I'm afraid I don't have much more time to put into this answer right now, but I suggest you read some introductory Objective-C, focusing on the difference between objects and classes, and the lifetime of objects and memory management. Essentially, nothing is ever persisted. No matter how many times you call -pickACard on any instance of DeckOfCards, you will always get a separate instance instantiated which will then get a brand new NSMutableArray instantiated. But none of these objects are ever retained (assuming you're using Automatic Reference Counting) and they all disappear immediately. So you will always, every time, get a brand new 52 card deck from -pickACard.

Hopefully this helps. With some quality time put into reading and learning the fundamentals of Objective-C and Object Oriented Programming, you will be well on your way to making this program work like you want. Let me know if you have questions and I'll try to answer them.

As the other commenters have said, you have multiple problems with the structure of your app. I've created an example app to show you how to start approaching this problem. First, in your DeckOfCards class you want a property that points to the mutable array that holds your deck of cards -- I called this just deck. Then, you need an init method that not only returns an instance of your DeckOFCards class, but also creates and populates that array of cards. So here is what I put in the DeckOfCards class.

In the .h, just this:

@interface DeckOfCards : NSObject

@property (retain,nonatomic) NSMutableArray *deck;

@end

And, in the .m :

-(id)init {
    if (self = [super init]) {
        NSArray *cardNames = [NSArray arrayWithObjects:@"Two",@"Three",@"Four",@"Five",@"Six",@"Seven",@"Eight",@"Nine",@"Ten",@"Jack",@"Queen",@"King",@"Ace",nil];
        NSArray *suits = [NSArray arrayWithObjects:@" of Hearts",@" of Diamonds",@" of Spades",@" of Clubs",nil];
        self.deck = [NSMutableArray array];
        for (NSString *aCard in cardNames) {
            for (NSString *aSuit in suits) {
                [self.deck addObject:[aCard stringByAppendingString:aSuit]];
            }
        }
    }
    return self;
}

You want to create an instance of this class, and pick cards from inside another class, not from within the DeckOfCards class itself. In this example I've done that in the app delegate (for simplicity ), but it would be better to do this in some controller class. So, I have a property, aNewDeck to keep a reference to that instance (the compiler won't let you use a name that starts with "new", not sure why, or if that's a new thing in mountain lion). I create that new instance in the applicationDidFinishLoading method, and then have an IBAction connected to a button that picks the card. Notice that I changed the random number generator to not use 52, but the count of the array -- if you don't do that, you'll be picking numbers that are bigger than the array as it gets smaller. You should notice, that to get access to the array, I use aNewDeck.deck which is the way to access that property, deck, of the instance aNewDeck. Here's the .h file:

@class DeckOfCards;
#import <Cocoa/Cocoa.h>

@interface AppDelegate : NSObject <NSApplicationDelegate>

@property (assign) IBOutlet NSWindow *window;
@property (strong) DeckOfCards *aNewDeck;

-(IBAction)pickACard:(id)sender ;

@end

An here's the .m:

#import "AppDelegate.h"
#import "DeckOfCards.h"

@implementation AppDelegate

- (void)applicationDidFinishLaunching:(NSNotification *)aNotification {
    self.aNewDeck = [[DeckOfCards alloc] init];
}

-(IBAction)pickACard:(id)sender {
    if (self.aNewDeck.deck.count != 0) {
        int r = arc4random() % [self.aNewDeck.deck count];
        NSLog (@"The card you picked is: %@, and there are %li cards left", [self.aNewDeck.deck  objectAtIndex:r], [self.aNewDeck.deck count] - 1 );
        [self.aNewDeck.deck removeObjectAtIndex:r];
    }else{
        NSLog(@"Game Over!");
    }
}

I hope this helps you get started.

DeckOfCards is not a subclass of NSMutableArray and it doesn't implement the - objectAtIndex: and - removeObjectAtIndex: methods either - so it's fine that you can't use these methods.