Next: Exponential Generating functions Recurrence relations P13, P112(Sixth) OR P13, P100(Fifth) Exercise: 1. Determine the number of 12-combinations of the multiset S={4·a,3·b,5·c,4d} 2. Determine the number of solutions of the equation x,+x,+x3+x4=14 in nonnegative integers X1x, X3, and xa not exceeding 8 3. Determine the number of permutations of 1, 2, 3, 4, 5, 6, 7, 8 in which no even integer is in its natural position 4. Determine the number of permutations of (1, 2,., n in which exactly k integers are in their natural positions. 5. Eight boys are seated around a carousel. In how many ways can they change seats so that each has a different boy in front of him? 6. Let s be the multiset{o·e1,o·e2y…,o·ek}. Determine the generating unction for the sequence ao, a1,.an i where a is the number of n- combinations of s with the added restriction 1)Each e; occurs an odd number of times. 2)the element e, does not occur, and e, occurs at most once. 7. Determine the generating function for the number an of nonnegative integral solutions of 2e1+5er+e3+7e=n▪ Next: Exponential Generating functions; ▪ Recurrence Relations P13, P112 (Sixth) OR P13,P100(Fifth) ▪ Exercise: 1. Determine the number of 12-combinations of the multiset S={4·a,3·b,5·c, 4·d }. ▪ 2. Determine the number of solutions of the equation x1+x2+x3+x4=14 in nonnegative integers x1 ,x2 ,x3 , and x4 not exceeding 8. ▪ 3.Determine the number of permutations of {1,2,3,4,5,6,7,8} in which no even integer is in its natural position. ▪ 4.Determine the number of permutations of {1,2,…,n} in which exactly k integers are in their natural positions. ▪ 5.Eight boys are seated around a carousel. In how many ways can they change seats so that each has a different boy in front of him? ▪ 6.Let S be the multiset {·e1 ,·e2 ,…, ·ek }. Determine the generating function for the sequence a0 , a1 , …,an , … where an is the number of n￾combinations of S with the added restriction: ▪ 1) Each ei occurs an odd number of times. ▪ 2) the element e2 does not occur, and e1 occurs at most once. ▪ 7.Determine the generating function for the number an of nonnegative integral solutions of 2e1+5e2+e3+7e4=n