Question

A pennant is a sequence of numbers, each number being 1 or 2. An n-pennant is a sequence of numbers with sum equal to n. For example, (1,1,2) is a 4-pennants. The set of all possible 1-pennant is {(1)}, the set of all possible 2-pennants is {(2), (1,1)} and the set of all 3-pennants is {(2,1), (1,1,1), (1,2)}. Note that the pennant (1,2) is not the same as the pennant (2,1). The number of 10-pennants is

• A. 89

Answer 1

The correct option is A 89
In a 10-pennat let there we $x_1$ ones and $x_2$ twos.
So we need to find all the solution of
$x_1$ + 2$x_2$ = 10
Put, $x_1$ = 0 $\Rightarrow$ $x_2$ = $\frac{10}{2}$ = 5
So, (0, 5) is a solution i.e. a 10-pennant could have 0 ones and 5 twos. The number of ordered permutations of 0 ones and 5 twos = 5! / 5! = 1
Now $x_1$ cannot be 1 since in that case $x_2$ = $\frac{9}{2}$ = 4.5(is not an integer).

Put, $x_1$ = 2 $\Rightarrow$ $x_2$ = $\frac{8}{2}$ = 4
So, (2, 4) is a solution i.e. a 10-pennant could have 2 ones and 4 twos. The number of ordered permutations of 2 ones and 4 twos

= $\frac{6!}{2!4!}$ = 15

Similarly (4, 3), (6,2), (8, 1) and (10, 0) are the other four solution and the number of pennants for each is respectively

$\frac{7!}{4!3!}$ = 35, $\frac{8!}{6!2!}$ = 28,

$\frac{9!}{8!1!}$ = 9, $\frac{10!}{10!0!}$ = 1

So, the total number of 10-pennants = 1 + 15 + 35 + 28 + 9 + 1 = 89