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The number of COMBINATIONS of $n$ distinct objects taken $r$ at a time, denoted $C(n,r)$ or $\binom{n}{r}$, is
A$\dfrac{n!}{r!\,(n-r)!}$
B$\dfrac{n!}{(n-r)!}$
C$\dfrac{n!}{r!}$
D$n \cdot r$
Answer & Solution
Correct answer: A. $\dfrac{n!}{r!\,(n-r)!}$
1. NCERT §6.4 (Combinations): a combination is an UNORDERED selection.
2. Each selection of $r$ objects can be permuted in $r!$ ways, but these all count as the SAME combination.
3. So $C(n, r) = P(n, r)/r! = \dfrac{n!}{r!(n-r)!}$.
4. Example: $C(5, 3) = 10$ (matches $P(5,3) = 60$ divided by $3! = 6$).
5. KEY relation: $P(n,r) = C(n,r) \cdot r!$.
6. Symmetry: $C(n, r) = C(n, n-r)$.
7. Option B is permutation. Option C is also permutation. Option D is wrong.
_Source: NCERT Class 11 Mathematics, Ch 6, §6.4 (Combinations — Eq. 6.9), p. 9–11._
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