How many 3-character PINs exist using digits 0-9 (repetition allowed)?
That's P(10,3)/something; repetition is allowed.
Correct — each position has 10 independent choices; product rule gives 10³ = 1000.
720 = P(10,3) = 10·9·8 — that's without repetition.
120 = 5! — unrelated here.
Recap: repetition allowed + sequential choices = product rule. 10 × 10 × 10 = 10³ = 1000.
The key difference between P(n,r) and C(n,r) is
Both use factorials in their formulas.
Correct — same elements in different order count once in C, multiple times in P.
P(n,r) ≥ C(n,r) always holds (for r ≥ 2), but that's a consequence, not the definition.
Neither allows repetition in the standard formulas.
Recap: permutation = order matters (ABC ≠ BAC). Combination = order doesn't matter (ABC = BAC = CAB). C(n,r) = P(n,r) / r!
C(5, 2) equals
That's P(5,2) = 5·4 = 20.
Correct — C(5,2) = 5!/(2!·3!) = 120/12 = 10.
60 = P(5,3).
25 = 5² — unrelated.
Recap: C(5,2) = 5·4 / 2·1 = 20/2 = 10. Always divide by r! to remove ordering from P(n,r).
Inclusion-exclusion for |A ∪ B| corrects for
Elements outside both sets aren't in A ∪ B at all.
Correct — |A| + |B| double-counts |A ∩ B|; we subtract it once.
Inclusion-exclusion counts sizes, not orderings.
Empty set contributions are zero; that's not what's corrected.
Recap: |A ∪ B| = |A| + |B| − |A ∩ B|. Without subtracting the overlap, elements in both sets are counted twice.
The coefficient of x²y in (x+y)³ is
The coefficient of x³ is 1; x²y has a larger coefficient.
Correct — C(3,1) = 3. The term is C(3,1)x²y¹ = 3x²y.
6 would be C(4,2); here n=3 and k=1.
9 = 3² — unrelated.
Recap: (x+y)³ = x³ + 3x²y + 3xy² + y³. The coefficient of xn−kyk is C(n,k).