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Tamattu` is the best type of Hajj. It is the one that Prophet Muhammad (peace and blessings be upon him) enjoined upon his Companions, and he wished that he had performed it.

Answering this question, Sheikh M. S. Al-Munajjid, a prominent Saudi lecturer and author, stated,

If someone performs `Umrah in Shawwal and stays in Makkah and then performs Hajj in the same year, then their Hajj is Hajj of Tamattu` and they have to offer a sacrifice.

That is because Tamattu` means assuming ihram for `Umrah during the months of Hajj and finishing the `Umrah and then assuming ihram for Hajj in the same year. The months of Hajj include Shawwal, Dhul-Qi`dah, and the first 10 days of Dhul-Hijjah.

The scholars differ on the case in which such pilgrims travel during the period between `Umrah and Hajj. The correct view is that if the pilgrims go back to their own cities or countries, then they cancel Tamattu`. But if they travel to any place other than their own cities or countries, then their Tamattu` remains in effect, and if they perform Hajj in the same year, they must offer a sacrifice.

The scholars of the Standing Committee were asked, "I did the rituals of `Umrah in Shawwal of AH 1395, and after doing that, I went back to my own town, and I have decided, in sha' Allah, to perform Hajj this year, AH 1395. Will I have to offer a fidyah (ransom or sacrifice)?"

They replied,

The majority of fuqahaa' [Muslim jurists] are of the view that you do not have to offer a sacrifice, because you did not do Tamattu` by doing `Umrah and Hajj in one trip, as you said that you went back to your town after performing `Umrah in Shawwal of AH 1395, and you did not stay in Makkah until Hajj.

Some fuqahaa' are of the view that you do have to offer a sacrifice if you perform Hajj in the same year, even if you did go back to your own city or country. This is because of the general meaning of the words of Allah: (If any one wishes to continue the `Umrah on to the hajj, He must make an offering, such as he can afford)  (Al-Baqarah 2:196).  

However, fatwas and practices in this regard are based on the view of the majority: that it is not obligatory for this pilgrim to offer a sacrifice in that case.

Sheikh Muhammad ibn Al-`Uthaymeen (may Allah have mercy on him) said,

If someone is doing Tamattu`, they may travel between `Umrah and Hajj to their own country or elsewhere, but if they travel to their own country and then return assuming ihram for Hajj, then they cancel their Tamattu` and do Ifrad. If they travel to a place other than their city or country and come back in ihram for Hajj, then their Tamattu` is still valid and they have to offer a sacrifice, as is well known.

In conclusion, if you went back to your own country or city after doing `Umrah, then the Tamattu` was cancelled and you do not have to offer a sacrifice. If you stayed in Makkah or traveled to any city or country other than your own (such as Madinah), then your Tamattu` was not cancelled.

Now, if you went back to your own city or country and thus cancelled your Tamattu`, then it is better for you to assume ihram for `Umrah at the Miqat when you travel for Hajj, so that you will be doing Tamattu`.

It should also be noted that the hadi (sacrifice) is not a penalty that people should try to avoid; rather, it is an act of worship by means of which Muslims draw closer to Allah, Exalted, and by means of which they increase their piety and righteous deeds and thank Allah for enabling them to perform the rituals [of Hajj and `Umrah] in one trip. And Allah knows best.

Excerpted with modifications from www.islam-qa.com.