The First Sikkim War was a massive war that encompassed most of Asia and nearly led to WW-III. It is also called the Second Sino-Indian War and the First Sino-Russian War.
China was at this period in an expansionist phase. Moreover, they were extremely confident in the war, having steamrolled Indian troops in the First Sino-Indian War. Meanwhile, their initial outnumbering of Russia on the Northern border led to a general feeling that the war would end swiftly. Mao and several others of the high Party leaders also hoped for a victory, which would help stem the massive discontent rising from Mao's unpopular Cultural Revolution and it's purges. Chinese army readiness at this oint, however, was different. The generals were split over the war; some believed that the Chinese army would still be outnumbered, despite it's size, losing their number one advantage, while also being poorly equipped and armed against their stronger Soviet neighbors.
Most generals, however, believed that the Himalayas would prevent Indian troops from breaking through into Tibet, while Soviet troops could be bogged down with fighting in the Palmirs and Kamchatka, and thus be defeated there, despite the state of the army.
India was, too, in a tense time. With their massive defeat against China in 1962 and then a meaningless war of attrition with Pakistan in 1965, the nation was discontented, especially with its slow economic growth as a result of very leftist leaning policies instituted by the government.
The Indian military had three main advantages- firstly, it's equipment was up to par with the Soviets, since it was mostly supplied by them; secondly, it had a large and well organized army; and thirdly, it had an excellent defensive position in Sikkim. The military were confident that, if unable to effect a victory, would at least be able to tie down enough Chinese troops to allow the Soviets to break into Beijing.
The USSR was also, militarily, ready for war, having the best equipment of the three sides and over half a million troops on the border. In addition, the government warned that they were ready and willing to use nuclear weapons in the event that China detonated their limited stock.
However, the people did not want such a war, that would exhaust the Union even further of resources. Public discontent rose sharply after this war.
The ROC was on the verge of losing its position on the UNSC. Desperate for a victory over the PRC, to regain some status, they saw this war as the best way to it, though they were unable to contribute much to the war, merely taking over after the Summer Coup destabilized the PRC.
The ROC's military, though not large, was certainly capable, being aided by, among many other nations, the USA.
Early Phases : August- November, 1968
The war started on August 1st, after Russia and India declared war on China after the latter nation set a deadline for the Soviet removal of all troops in the Palmirs.