The three problems that the newly independent nation of India faced are as follows:

1) Rehabilitation of a large number of refugees.
2) Assimilation of princely states.
3) Ensuring the unity of a country which is full of diversity.

Five problems faced by newly independent India are:

1. Partition and Communal Tensions
- India was divided into two countries, India and Pakistan, which resulted in displacement and violence.
- Communal tensions between Hindus and Muslims were high, resulting in riots and conflicts.

2. Economic Challenges
- India faced economic challenges due to the drain of resources during the colonial era.
- The country had to develop its infrastructure and industries to provide employment and improve the standard of living.

3. Illiteracy and Education
- India had a high illiteracy rate, with only a small percentage of the population having access to education.
- The government had to focus on improving the education system and providing educational opportunities to all citizens.

4. Caste System and Discrimination
- The caste system was deeply ingrained in Indian society, which resulted in discrimination against lower castes and women.
- The government had to work towards eliminating discrimination and promoting equality.

5. Political Instability
- India had a diverse population with different languages, religions, and cultural practices, which made it difficult to govern.
- Political instability was common, with frequent changes in government and political parties.

Overall, India faced numerous challenges after gaining independence, but the government worked towards addressing these issues and promoting growth and development.