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 ______. [2014]

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 ______. [2014]

Smart Achievers · Accepted Answer

(7)

$∵$ $n_{1}, n_{2}, n_{3}, n_{4} and n_{5} are positive integers such that n_{1} < n_{2} < n_{3} < n_{4} < n_{5}$

$Then for n_{1} + n_{2} + n_{3} + n_{4} + n_{5} = 20$

$If n_{1}, n_{2}, n_{3}, n_{4} take minimum values 1, 2, 3, 4 respectively, then n_{5} will be maximum 10 .$

$∴$ $Corresponding to n_{5} = 10, there is only one solution$

$n_{1} = 1, n_{2} = 2, n_{3} = 3, n_{4} = 4 .$

$Corresponding to n_{5} = 9, we can have only one solution$

$n_{1} = 1, n_{2} = 2, n_{3} = 3, n_{4} = 5 i.e., one solution.$

$Corresponding to n_{5} = 8, we can have, only solution$

$n_{1} = 1, n_{2} = 2, n_{3} = 3, n_{4} = 6$

$or n_{1} = 1, n_{2} = 2, n_{3} = 4, n_{4} = 5$

$i.e., 2 solutions.$

$For n_{5} = 7, we can have$

$n_{1} = 1, n_{2} = 2, n_{3} = 4, n_{4} = 6$

$or n_{1} = 1, n_{2} = 3, n_{3} = 4, n_{4} = 5$

$i.e., 2 solutions.$

$For n_{5} = 6, we can have$

$n_{1} = 2, n_{2} = 3, n_{3} = 4, n_{4} = 5$

$i.e., one solution.$

$∴$ $Thus there can be 7 solutions.$

Solution

Q.
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 ______. [2014]

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Solution

Q. Let n1<n2<n3<n4<n5 be positive integers such that n1+n2+n3+n4+n5=20. Then the number of such distinct arrangements (n1,n2,n3,n4,n5) is ______. [2014]

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Q.
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 ______. [2014]