Question

Let `n_1<n_2<n_3<n_4<n_5` be positive integers such that `n_1+n_2+n_3+n_4+n_5=20.` then the number of such distinct arrangements `n_1, n_2, n_3, n_4, n_5` is

Answer 1

Correct Answer - 7
PLAN Reducing the equation to a newer equation, where sum of variables is less. Thus, finding the number of arrangements becomes easier.
As `n_(1) ge 1, n_(2) ge 2, n_(3) ge 3, n_(4) ge 4 n_(5) ge 5`
Let `n_(1) - 1 = x_(1) ge 0, n_(2) - 2 = x_(2) ge 0, ..., n_(5) - 5 = x_(5) ge 0`
`rArr` New equation will be
`x_(1) + 1 + x_(2) + 2 + ... + x_(5) + 5 = 20`
`rArr " " x_(1) + x_92) + x_(3) + x_(4) + x_5) = 20 - 15 = 5`
Now, `x_(1) le x_(2) le x_(3) le x_(4) le x_(5)`

So, 7 possible cases will be there.