Question

Please help me with #6 a) and b)

Book: Applied combinatorics by Fred Roberts

6. (Cohen [1978]) (a) In a six-cylinder engine, the even-numbered cylinders are on the left and the odd-numbered cylinders are on the right. A good firing order is a permutation of the numbers 1 to 6 in which right and left sides are alternated. How many possible good firing orders are there which start with a left cylinder? (b) Repeat for a 2n-cylinder engine. the

Book answer (picture below)

Answer 1

Give data, In a six-cylinder engine the numbered cylinder are not be and on the left and add on the right. A good firing order is a permutation of number 1 to 6 which the sides of the engine is alternate. A, if a firing starts ion right. Implies then we will have 3 choices to the first cylinder. Implies now on left fire then fire second of cylinder. Implies now, we are left with two choices it will be go on. Implies and then we can use multiplication principle. And we get 3*2*2*2*1 = 36 so, number of possible good firing orders are there starting with the first cylinder is 36. Let us consider cylinder left stay left and it right. And we are given a 2n-cylinder engine. Implies so, we can arrange with n! possible ways. Implies let us decide with odd numbers. Implies so, we will get n! * n! ways. Implies now, we have 2n cylinder. Implies so, the