Such coalitions in which cabinet responsibilities are not shared are called legislative coalitions. Parties that agree to support the cabinet without getting cabinet representation are referred to as support parties. The more common type of coalitions, however, are the one that all the members want to share the control of the executive branch – known as cabinet or executive coalitions. 2 Politics view
Coalition Theories:
Cabinets have to be formed so that they have the confidence of the parliamentary majority. Several theories have been proposed to predict what coalitions would form in parliamentary systems:

Parties A(left) B C D E(right)
Seats 8 21 26 12 33

Theories :

Minimal winning coalition ABC ADE BCD BE CE
Minimal size ADE
Bargaining position BE CE
Minimal range ABC BCD CE
Minimal connected winning ABC BCD CDE
Policy-viable coalition ABC BCD CE
1) Minimal winning coalitions:

This theory was predicted by William H. Riker who, based on his “size criterion ”, states that only those parties will unite that are minimally necessary to win the majority in the parliament. Thus a coalition between parties ABC is one of the likely to be formed because the elimination of one member would leave it with a minority and also the addition of another member would make it larger than minimal.
The assumption lying under this theory is that parties are interested in maximizing their power, so they aim to have as many cabinet seats as possible. That is why to enter the cabinet a minority party would be willing to be a part of a coalition with another party, but each of them would not want to welcome more parties than needed in the coalition because that would reduce their own share of seats in cabinet.
When there is a majority party in the parliament this theory predicts only one outcome – non-coalition cabinet formed by the majority party. If there is no majority party the resulting coalitions might be many. In the example with five parties in the table above the coalitions likely to be formed are five.
Riker’s theory also assumes that parties have full information while they bargain; that prize for the winning coalition is the same regardless of who its members are; and that each case of coalition bargaining is an independent event, in which parties do not pay attention to what has happened in the past or what they expect to happen in the future. Critiques challenge Riker’s assumption that parties are totally indiscriminate in their search for partners.