Home › JEE Main › Mathematics › Permutations and Combinations › $^{n}C_r$, the number of combinations of $n$ dis…
$^{n}C_r$, the number of combinations of $n$ distinct objects taken $r$ at a time, equals:
A$n + r$
B$\dfrac{n!}{r!}$
C$\dfrac{n!}{r!(n-r)!}$
D$\dfrac{n!}{(n-r)!}$
Answer & Solution
Correct answer: C. $\dfrac{n!}{r!(n-r)!}$
$^{n}C_r = \dfrac{n!}{r!(n-r)!}$.
Useful identity: $^{n}C_r = \dfrac{^{n}P_r}{r!}$ — divide permutations by $r!$ because every selection of $r$ objects can be arranged $r!$ different ways, and combinations ignore order.
For $r > n$, $^{n}C_r = 0$ (can't choose more than you have). Symmetry: $^{n}C_r = {^{n}C_{n-r}}$.
Related questions
Number of ways to choose 2 items out of 5:Number of permutations of letters in WORD:How many DIAGONALS does a regular DECAGON (10-sided polygon) have?$\binom{n}{r} + \binom{n}{r-1}$ equalsFind the number of ways a person can travel from city A to city C if there are 3 routes frHow many distinct 'numbers' (with REPETITION ALLOWED) can be formed using the digits 0, 1,The number of ways to select 2 boys and 3 girls from a group of 5 boys and 4 girls isHow many distinct 4-digit numbers can be formed using the digits 1, 2, 3, 4, 5, 6 (each di