When six fair coins are tossed simultaneously, in how many of the outcomes will at most three of the coins turn up as heads?

The question requires you to find number of the outcomes in which at most 3 coins turn up as heads.
i.e., 0 coins turn heads or 1 coin turns head or 2 coins turn heads or 3 coins turn heads.
The number of outcomes in which 0 coins turn heads is,
⁶C₀ = 1 outcome.
The number of outcomes in which 1 coin turns head is,
⁶C₁ = 6 outcomes.
The number of outcomes in which 2 coins turn heads is,
⁶C₂ = 15 outcomes.
The number of outcomes in which 3 coins turn heads is,
⁶C₃ = 20 outcomes.
Therefore, total number of outcomes
= 1 + 6 + 15 + 20
= 42 outcomes.