There are many variations on the basic defense outlined so far. We'll look briefly at the most common variations, the nickle and dime packages, the prevent defense, and the 3-4.
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Some teams like to pass the ball a lot. If a team has a good quarterback and three very good wide receivers, it would be foolish for them to leave a receiver on the bench just because "normal" football alignments include a fullback. So such a team will frequently take their fullback off the field and substitute a third wide receiver, a slot receiver. This is especially likely if the situation on the field calls for a pass, for example it's third down with 8 yards to go. Few teams would run the ball in this situation.
When a fast wide receiver is put in place for a fullback, there is a mismatch on the field. You might expect that the weak side linebacker would cover the third wide receiver, but there is little chance he can successfully do this. A linebacker is a 245 pound guy who is trained to deliver big hits, and the third wide receiver is most likely a 185 pound guy who runs like the wind. The receiver is going to run right by the linebacker. So the defense will most likely remove the weak side linebacker from the field and substitute a third cornerback. The defense now has three cornerbacks and two safeties on the field, for a total of five defensive backs. Since there are five backs, this is called a nickle package. If you are defending against a passing team like the Rams or the Colts, it is likely that your team will start out the game in their nickle package and never actually play their base defense. Of course it's very important that your team have a quality third cornerback, or you will still have a mismatch on the field and you are still in trouble.
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There are several running backs in the NFL who are just as accomplished catching the ball as they are running with the ball. In obvious passing situations, many teams will put their running back into motion or split him out as a fourth wide receiver, or in a few cases if the team has four quality wide receivers they might even remove their running back from the game and substitute a fourth wide receiver. In any case, you now have another case where a fast guy, a wide receiver or running back, is being covered by a linebacker. This is again an obvious mismatch, and the defense is in trouble. So the defense will respond by pulling another linebacker off the field and substituting a fourth cornerback. There are now six defensive backs on the field. Just as five defensive backs is called the nickle package, six defensive backs is called a dime package. If your team doesn't have a quality fourth cornerback, you're in trouble.
In the nickle and dime packages, there are few linebackers on the field, so the defense is more vulnerable to a running play. The idea is that if it's third down with eight or more yards to go, the safeties can get up and stop the running back before he gets eight yards. On first down this would be a terrible blow to the defense, to give up seven yards on a running play. But on third down with eight yards to go, giving up seven yards means the defense wins and gets to go sit down.
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In the quarter package, the defense will pull another fat guy off the field, likely a tackle, and replace him with a seventh defensive back. In a quarter package it's a near certainty that the offense can successfully run the ball for four to eight yards per carry. So you would only use this package when you would be happy for the offense to run the ball a lot, for example when there is very little time left in the game and you are winning by 17 or more points. The quarter package is used to prevent quick scores and make the offense waste a lot of time.