Question

10. (a) By giving a combinatorial argument show that [mk=n-21-1 (Hint: Think of ways of choosing a committee and its chairperson.) (b) Find a closed form formula for į ()k. (That is, a formula that does not involve summation with varying number of terms. For instance, Part (a) gives a closed form formula for

Answer 1

of members and we've to suppose we a team E fix some boden of it have a Subfamily of it of leader. choose choose the Size, and any be imclude d into Then, Probakility of the Subfamily of member to gach (osksm) k members k members' Subfamily of Theus, chanceé of formming Specifie family leader. for this and P(x=k) to fix themm choices we're K. many a farmily of k Thus, Average /Expected value of forming a family of k مصله E k. P(x =K) k-o . ΣΚ. probability asso ciated But Since to each nen betng n men, the above probabítity selected =b,amengst actually is

.)ع( و و ۴=0 .2حا د -1) ! إ)-ج ۰ سة K, -K)! k! (n-k)! (k-) ! kzo ( و ([ - (() Σ %3D (using prenious reult 2O-). - na+) 2^-* + 2^* ۰ ۸ه سرام م ع اما ار ا )و