The ATX +12v 4-pin connector does not plug into the 24 pin connector on your MB … (it wouldn't fit anyway as the pin configuration is different). There is a separate 4-pin connector on the MB where the ATX +12v 4-pin connector goes to provide extra power for the CPU. On newer MB's it's an 8-pin EPS connector.
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You can plug a 20-pin ATX into a ATX 24-pin on your MB, it's pin compatible, and it will work without the extra 4 pins. The extra 4pins added to the ATX 20-pin connector to make the 24 ATX includes an additional +12v pin (and a 3/5v pin) because the original 20-pin ATX only has one +12v pin on it. As newer CPU's came out in the early 2000's they moved towards using more 12v power and less (3/5v), and more power was needed on the +12v connection. Both +12v pins feed the same power rail on the MB, which is why it works.
One caution though. A single molex connector pin is rated for 6A (amps) x 12V = 72W (watts). With the new high powered CPU's, this may not be enough in some cases, and one would be pushing more than what the single 12v molex pin is rated for. ( the two +12v pins of the ATX 24-pin can provide double -- 144 watts total).
In this
xbit test, a dual core E8600, 3.3Ghz, + RD 4850 graphics card managed to pull just under 4A peak (about 45 watts) from the 24-pin ATX 12v connection. So in this case a PSU with just a ATX 20-pin connector would suffice with room to spare. But if one did some extreme overclocking and overvolted the stock CPU, one could probably get close to the 6A (72 watt) limit for just one molex 12v pin on the 20 pin ATX connector for this system.
On the quad core system, i7-920, the maximum draw from the ATX 24-pin connector on the 12v lines was over 72 watts, which means a 20-pin ATX connector with only one 12v pin would not be sufficient.
As mentioned, you can get a 24-pin adapter, but it still means that only 1 +12v pin (split into 2 feeds) is powering the MB. So you still have the 72 watt limit and probably wouldn't want to power a quad core system with an 20 - 24 pin ATX adapter.