The sacrifice (هدي) is واجب for الحجاج performing الحج al-Tamattu and الحج al-Qiran, based on القرآنic verse: 'And whoever enjoys العمرة followed by الحج must offer what is easily available of sacrificial animal. But whoever cannot find (one) — then a fast of three days during الحج and seven when you return home. That is ten complete days' (2:196). This verse establishes both the obligation of sacrifice and the alternative of fasting for those unable to afford it.
الشيخ ابن باز explained that the sacrifice is a gratitude offering (shukr) for the blessing of combining العمرة and الحج in one journey, saving the الحاج the cost and effort of separate trips. The minimum acceptable sacrifice is one sheep or goat that meets the age requirements (six months for sheep, one year for goats). Alternatively, a الحاج may share in a one-seventh portion of a camel or cow (seven الحجاج sharing one camel or cow). The animal must be free of defects that would disqualify it.
الشيخ ابن عثيمين clarified that الحجاج performing الحج al-Ifrad (الحج alone without العمرة) are not required to offer a sacrifice, as they did not combine two acts of worship. He also noted that residents of مكة who perform Tamattu are not required to offer a sacrifice عند جمهور العلماء, based on the continuation of the verse: 'This is for those whose family is not present at al-المسجد الحرام.' The sacrifice obligation applies to non-مكة residents who perform Tamattu or Qiran.