Answer:

Introduction      

The Indian Constitution is written, codified, and regarded as the supreme law of the land. It is the world’s longest constitution with around 470 Articles (divided into 25 Parts) and 12 Schedules. The Indian Constitution uniquely amalgamates both rigidity and flexibility, ensuring stability while permitting essential evolutionary changes.

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Ways in which the Indian Constitution incorporates elements of both rigidity and flexibility 

Rigidity in the Indian Constitution:

• Amendment of Federal Provisions Article 368(2): Any amendment that affects the federal structure of the Constitution, like the powers and functions of states, requires not just a two-thirds majority in both houses of Parliament, but also the ratification by legislatures of at least half of the states. 
• Basic Structure: Established by the landmark Kesavananda Bharati case in 1973, it ensures certain core features of the Constitution remain unalterable. This includes the Parliamentary system, Judicial Independence, and Fundamental Rights, thereby placing defined constraints on the Parliament’s amending powers.
• Protection of Fundamental Rights: Fundamental Rights (Article 14-32) can only be amended by the special majority, that involves a majority of 2/3rd members present and voting supported by over 50% of the total strength of the House. Further, any amendment in the Fundamental Rights can be reversed by the judiciary on the grounds of violation of the Basic Structure of the constitution. 
• Representation of States in Parliament (Article 80): The representation of states in Parliament, especially in the Rajya Sabha (Schedule 4) is based on a formula that cannot be easily altered, ensuring a degree of rigidity in federal representation.

Flexibility in the Indian Constitution:

• Simple Majority Amendments: Article 4 of the Indian Constitution stipulates that certain constitutional provisions can be amended by simple majority of the two Houses of Parliament, just like any ordinary law. Eg: the creation of a new state, alteration of state boundaries, require only a simple parliamentary majority.
• DPSPs (Part IV): They serve as  ‘Instrument of instructions‘ for good governance. They can be updated or changed to reflect contemporary requirements. Eg: 44th Amendment 1978 created a new DPSP under Article 38, which mandates the State to reduce inequalities in income, status, facilities, and opportunities.
• Adaptable Executive and Legislative Powers: The powers of the President, as laid out in the Constitution, are broad and can be exercised flexibly, subject to the Parliament’s will. Similarly, the Parliament under Article 248 (residuary powers) can legislate on matters not enumerated in any list. 
• Adoption of Global Treaties: The Indian Parliament, under Article 253, can seamlessly incorporate international agreements. The Paris Agreement on climate change, ratified by India in 2016, showcases this flexible integration.

Different methods through which amendments to the Constitution can be made

The process of amending the Constitution of India is defined in Article 368 of Part XX. The second and third types of amendments are governed by Article 368. The Indian Constitution can be amended in three ways: 

• Simple Majority: As per Article 2 and 3, certain changes like the admission or establishment of new states, alteration of areas, boundaries, or names of existing states require only a simple majority in both Houses of Parliament. Eg: the renaming of the state of Orissa to Odisha in 2011 was executed by a simple parliamentary majority.
• Special Majority (Article 368): It involves a two-thirds majority of members present and voting in each House of Parliament and absolute majority of its total membership, is indispensable. It is mainly required for amending FRs and DPSPs. Eg: 86th Amendment of 2002 ensuring the right to education under Article 21A.
• Special Majority with Half of State Ratification Article 368(2): It is required for changes that might alter the relationship between the Centre and the States. It requires a two-thirds majority in Parliament and ratification from half of the states. Eg: ratification of GST bill 101st amendment constitution by States.

Conclusion

The Indian Constitution’s beauty lies in its adept combination of rigidity and flexibility. While safeguarding core principles, it remains adaptable to the nation’s evolving needs. This dynamism, coupled with a comprehensive amendment procedure, ensures that India’s constitutional framework remains relevant, effective, and reflective of its democratic ethos.