The peak that lies between Nepal and Sikkim, with three out of five peaks directly on the border, is the Kanchenjunga.

Kanchenjunga is the third highest mountain in the world, with an elevation of 8,586 meters (28,169 feet). It is located in the eastern Himalayas, on the border between Nepal and the Indian state of Sikkim.

Here are the reasons why Kanchenjunga is the correct answer:

1. Geographical Location:
- Kanchenjunga lies in the Himalayan range, which stretches across several countries including Nepal, India, Bhutan, Tibet (China), and Pakistan.
- The mountain is situated specifically on the border between Nepal and Sikkim, which is a northeastern state of India.
- The peak is part of the Kanchenjunga Conservation Area, a protected area that spans both Nepal and Sikkim.

2. Border Peaks:
- Out of the five main peaks of Kanchenjunga, three of them lie directly on the border between Nepal and Sikkim.
- The main summit, Kanchenjunga Main, is the highest point and lies entirely in Nepal.
- The other four peaks are named Kanchenjunga West, Kanchenjunga Central, Kanchenjunga South, and Kangbachen. These peaks are spread along the border, with three of them directly on the border.

3. Cultural Significance:
- Kanchenjunga holds great cultural significance for the people living in Nepal and Sikkim.
- It is considered a sacred mountain in both Hinduism and Buddhism, and it is worshipped by the local communities.
- The mountain and its surrounding areas are also home to various indigenous communities with unique cultures and traditions.

In conclusion, Kanchenjunga is the peak that lies between Nepal and Sikkim, with three out of five peaks directly on the border. Its location, border peaks, and cultural significance make it a significant landmark in the region.