All private consumption in a nation's economy is part of this expenditures approach for calculating the GDP.

Answer: Yes, private consumption in an economy is a part of expenditure mthod approach to calculate GDP.

But this is a necessary but not sufficicent condition to calculate GDP by this method

Explanation:

GDPmp (by expenditure method) = Private Final Consumption Expenditure + Government Final Consumption Expenditure + Gross Domestic Capital Formation + Net Exports (Exports - Imports)

The method is based on circular flow of income notion 'one person's expenditure is other person's income'. Hence summing expenditure of all the sectors of economy (households, firms, govt, financial sector, rest of world) - we get the income of the entire economy.

Considering 'private final consumption expenditure':

This is = Households expenditure + expenditure of non public institutions serving households.

In tricky questions household expenditure might not be directly given, then it can be calculated by : Household Income - Household Saving