Binomial coefficients
86 problems · Binomial expansion, Pascal's triangle and combinatorial identities.
Ex.215Definition and basic values
Binomial coefficients
High school
Ex.216Pascal's triangle
Binomial coefficients
High school
Ex.217Symmetry of binomial coefficients
Binomial coefficients
High school
Ex.315Sum of binomial coefficients
Binomial coefficients
L1
Ex.330Pascal's rule
Binomial coefficients
L1
Ex.400Alternating sum of binomial coefficients
Binomial coefficients
High school
Ex.401The binomial theorem
Binomial coefficients
High school
Ex.402A specific expansion
Binomial coefficients
High school
Ex.403Key binomial identities
Binomial coefficients
High school
Ex.404Sum of binomial coefficients
Binomial coefficients
High school
Ex.405Monotonicity of binomial row
Binomial coefficients
High school
Ex.406Even vs odd subsets
Binomial coefficients
High school
Ex.407Pascal's identity
Binomial coefficients
High school
Ex.408Simplifying binomial quotients
Binomial coefficients
High school
Ex.409Recurrence for coefficients
Binomial coefficients
High school
Ex.410Binomial as a quotient of factorials
Binomial coefficients
High school
Ex.411Sum over odd $k$
Binomial coefficients
High school
Ex.412Sum over odd and even sizes
Binomial coefficients
High school
Ex.413Sum over even $k$
Binomial coefficients
High school
Ex.414Sum of powers of 2
Binomial coefficients
High school
Ex.415Sum with $3^k$
Binomial coefficients
High school
Ex.605A combinatorial proof that Σ C(n,k)² = C(2n,n)
Binomial coefficients
M1
Ex.799Parity of binomial coefficients
Binomial coefficients
L1
Ex.800A combinatorial proof
Binomial coefficients
Olympiad
Ex.801Combinatorial proof
Binomial coefficients
High school
Ex.802Compositions via binomial coefficients
Binomial coefficients
Olympiad
Ex.803Expansion of a large binomial
Binomial coefficients
High school
Ex.804Telescoping with binomials
Binomial coefficients
Olympiad
Ex.805The hockey-stick identity
Binomial coefficients
L1
Ex.806An inequality with binomials
Binomial coefficients
L1
Ex.807Counting with binomials
Binomial coefficients
L1
Ex.808The largest binomial coefficient
Binomial coefficients
L1
Ex.809Multinomial identity
Binomial coefficients
Olympiad
Ex.810Generalized binomial coefficients
Binomial coefficients
L1
Ex.811Polynomial expansion with binomials
Binomial coefficients
L1
Ex.812Coefficients of a power
Binomial coefficients
L1
Ex.813Application to sums
Binomial coefficients
L1
Ex.814Sum with two powers
Binomial coefficients
L1
Ex.815Vandermonde's identity
Binomial coefficients
L1
Ex.816Weighted sum of binomial coefficients
Binomial coefficients
L1
Ex.1308Vandermonde's identity and a weighted sum of squares
Binomial coefficients
M1
Ex.1526Binomial coefficient inverses
Binomial coefficients
Olympiad
Ex.1527An identity with inverses
Binomial coefficients
L1
Ex.1528Catalan via binomial
Binomial coefficients
L1
Ex.1529Central binomial coefficients
Binomial coefficients
Olympiad
Ex.1530Choosing subsets of subsets
Binomial coefficients
L1
Ex.1531Binomial coefficients and primes
Binomial coefficients
Olympiad
Ex.1532A combinatorial sum
Binomial coefficients
Olympiad
Ex.1533A combinatorial sum with choice
Binomial coefficients
Olympiad
Ex.1534Cyclic identity with binomials
Binomial coefficients
Olympiad
Ex.1535Approximation via binomial theorem
Binomial coefficients
Olympiad
Ex.1536Fibonacci and binomials
Binomial coefficients
Olympiad
Ex.1537Generating function of binomial row
Binomial coefficients
L1
Ex.1538A complex binomial identity
Binomial coefficients
Olympiad
Ex.1539Sum of sines
Binomial coefficients
Olympiad
Ex.1540An identity with squares
Binomial coefficients
Olympiad
Ex.1541Application to integration
Binomial coefficients
L1
Ex.1542An integral identity
Binomial coefficients
L1
Ex.1543The multinomial theorem
Binomial coefficients
L1
Ex.1544Multinomial coefficients
Binomial coefficients
L1
Ex.1545Partial sums of binomials
Binomial coefficients
Olympiad
Ex.1546Hexagon identity
Binomial coefficients
Olympiad
Ex.1547Pentagon identity
Binomial coefficients
Olympiad
Ex.1548Weighted power sum
Binomial coefficients
Olympiad
Ex.1549Quadratic coefficient sum
Binomial coefficients
Olympiad
Ex.1550Moments of the binomial row
Binomial coefficients
L1
Ex.1551Sum of squares of binomial coefficients
Binomial coefficients
L1
Ex.1552Stirling numbers and binomials
Binomial coefficients
M1
Ex.1553Convolution of binomial rows
Binomial coefficients
Olympiad
Ex.1554Reciprocals of binomial coefficients
Binomial coefficients
Olympiad
Ex.1555Sum with $k^2\binom{n}{k}$
Binomial coefficients
L1
Ex.1556A generalized Vandermonde
Binomial coefficients
L1
Ex.1939Paths that never go below the axis: Catalan numbers
Binomial coefficients
Olympiad
Ex.1976Sperner's theorem and the LYM inequality
Binomial coefficients
Olympiad
Ex.2084Chu–Vandermonde for negative upper index
Binomial coefficients
M1
Ex.2085A hard combinatorial identity
Binomial coefficients
Olympiad
Ex.2086A correct challenge identity
Binomial coefficients
Olympiad
Ex.2087Gauss's binomial theorem
Binomial coefficients
M1
Ex.2088Generating function of central binomials
Binomial coefficients
M1
Ex.2089A challenging binomial sum
Binomial coefficients
Olympiad
Ex.2090A challenge identity
Binomial coefficients
Olympiad
Ex.2091Lucas's theorem
Binomial coefficients
M1
Ex.2092Mahonian numbers
Binomial coefficients
M1
Ex.2093Partial fractions with binomials
Binomial coefficients
M1
Ex.2094Riordan arrays (statement)
Binomial coefficients
M1
Ex.2095Zeilberger's algorithm (statement)
Binomial coefficients
M1
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