Live practice

Define the sample space and count each favourable outcome once.

The final value lies in [0,1]; complement pairs also sum to one, providing an independent check.

Form the required totals explicitly and divide by the full number of observations.

The variance is nonnegative and the reported standard deviation is its positive square root.

Translate every condition into an equation and retain exact values throughout the conic calculation.

The last line satisfies the original conditions and supplies every requested part.

Use the sequence definition or geometric-progression formula, keeping the first-term offset explicit.

The displayed substitutions cover every requested part and verify the result against the original data.

Proceed from the given expression and keep every transformation explicit.

The last line satisfies the required domain and establishes the result.

Each coordinate has two choices, so \(2^3=8\) triples:

\(\{(-1,-1,-1),(-1,-1,1),(-1,1,-1),(-1,1,1),(1,-1,-1),(1,-1,1),(1,1,-1),(1,1,1)\}\).

Reconnect the carbonyl carbons of ethanal and pentan-3-one with a double bond and remove both oxygens.

Apply lowest set of locants, alphabetical tie-breaking and priority of the alcohol suffix.

Remember that the O–O bond in a peroxide makes each peroxide oxygen −1. Apply the charge balance, then draw structures that identify peroxide oxygen in H₂SO₅ and bridging oxygen in dichromate.

Use Kp=Kc(RT)^Δn. For (i), Δn=1; for carbonate decomposition, Δn=1.