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

Answer: Option D

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.