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Permutations And Combinations

Class 11 Mathematics के इस tag से जुड़े questions। हर question के साथ chapter, topic, level और difficulty दी गई है।

MathematicsFor a positive integer n, which derivation gives \(\binom{n}{0}+\binom{n}{2}+\binom{n}{4}+\cdots=2^{n-1}\)?Class 11Permutations and CombinationsDerivations of formulas and their connectionsHard MathematicsWhat is the first correct step in simplifying \(\binom{12}{4}+\binom{12}{5}+\binom{13}{6}\)?Class 11Permutations and CombinationsDerivations of formulas and their connectionsHard MathematicsUsing Pascal's identity, how can \(\binom{n}{r}+\binom{n}{r+1}\) be written?Class 11Permutations and CombinationsDerivations of formulas and their connectionsHard MathematicsFor integers \(0\le r\le n\), given \({}^{n}P_r=r!\binom{n}{r}\), why is \({}^{n}P_r\ge\binom{n}{r}\)?Class 11Permutations and CombinationsDerivations of formulas and their connectionsHard MathematicsWriting \(\binom{n}{r}=\frac{n(n-1)\cdots(n-r+1)}{r!}\) shows which connection?Class 11Permutations and CombinationsDerivations of formulas and their connectionsHard MathematicsFor fixed \(n\ge2\) and integers \(0\le r<s\le n-1\), why is \({}^{n}P_s>{}^{n}P_r\)?Class 11Permutations and CombinationsDerivations of formulas and their connectionsHard MathematicsFor integers 0 ≤ r, s ≤ n, if \(\binom{n}{r}=\binom{n}{s}\) and \(r\ne s\), what relation must hold?Class 11Permutations and CombinationsDerivations of formulas and their connectionsHard MathematicsIf n is odd, which two binomial coefficients C(n,r) are largest?Class 11Permutations and CombinationsDerivations of formulas and their connectionsHard MathematicsIf n is even, which binomial coefficient C(n,r) is uniquely largest?Class 11Permutations and CombinationsDerivations of formulas and their connectionsHard MathematicsWhy do the binomial coefficients C(n,0), C(n,1), …, C(n,n) increase toward the middle and then decrease?Class 11Permutations and CombinationsDerivations of formulas and their connectionsHard MathematicsFive identical prizes are distributed among 8 students, with each student receiving at most one prize. How many distributions are possible?Class 11Permutations and CombinationsDerivations of formulas and their connectionsHard MathematicsFour identical prizes are distributed among 10 students, and a student may receive more than one. Which count gives the number of distributions?Class 11Permutations and CombinationsDerivations of formulas and their connectionsHard MathematicsFour different prizes are given to 10 students, with no student receiving more than one prize. How many distributions are possible?Class 11Permutations and CombinationsDerivations of formulas and their connectionsHard MathematicsFour different prizes are distributed among 10 students, and a student may receive more than one prize. How many distributions are possible?Class 11Permutations and CombinationsDerivations of formulas and their connectionsHard MathematicsIn which case would replacing P(n,r) with n^r give the wrong count?Class 11Permutations and CombinationsDerivations of formulas and their connectionsHard MathematicsWhen forming four-digit numbers from available digits that include 0, why must the restriction on the first digit be handled separately?Class 11Permutations and CombinationsDerivations of formulas and their connectionsHard MathematicsFour-digit numbers are formed using six available non-zero digits, with repetition allowed. Why is the count 6^4?Class 11Permutations and CombinationsDerivations of formulas and their connectionsHard MathematicsThere are n distinct objects and r specified, distinct positions to fill, with no object used more than once. Which formula counts the arrangements?Class 11Permutations and CombinationsDerivations of formulas and their connectionsHard MathematicsFor n distinct people, with n ≥ 3, how many round-table seatings have A and B adjacent?Class 11Permutations and CombinationsDerivations of formulas and their connectionsHard MathematicsWhy are there (n−1)! round-table seatings but n! row seatings for n distinct people?Class 11Permutations and CombinationsDerivations of formulas and their connectionsHard