grandes-ecoles 2012 QI.A.3

grandes-ecoles · France · centrale-maths1__pc Binomial Theorem (positive integer n) Prove a Binomial Identity or Inequality
Show that $\sum _ { k = 0 } ^ { n } k ( k - 1 ) \binom { n } { k } x ^ { k } ( 1 - x ) ^ { n - k } = n ( n - 1 ) x ^ { 2 }$. 
Show that $\sum _ { k = 0 } ^ { n } k ( k - 1 ) \binom { n } { k } x ^ { k } ( 1 - x ) ^ { n - k } = n ( n - 1 ) x ^ { 2 }$.
Paper Questions
QI.A.1 QI.A.2 QI.A.3 QI.A.4 QI.B.1 QI.B.2 QI.C.1 QI.C.2 QI.C.3 QI.C.4 QII.A.1 QII.A.2 QII.B.1 QII.B.2 QII.B.3 QII.B.4 QII.C.1 QII.C.2 QII.C.3 QII.D.1 QII.D.2 QII.D.3 QII.E.1 QII.E.2 QII.E.3 QII.E.4 QII.E.5 QII.E.6