The number of $$non-negative$$ solutions of $$x1+x2+x3+$$,…,$$+xn$$≤n (where$$ n is positive $$integer) is

Question

The number of $$non-negative$$ solutions of $$x1+x2+x3+$$,…,$$+xn$$≤n (where$$ n is positive $$integer) is

Infinity Learn · Accepted Answer

In general, we know thatFor the distribution $$equationx1+x2+x3+$$…$$+xn$$≤nLet required ways $$=$$ W⇒$$W=No$$. of ways of distributing $$1$$ $$item+No$$. of ways of distributing $$2$$ $$items+$$......$$+No$$. of Ways of distributing n $$items=1+n$$−$$1Cn$$−$$1+2+n$$−$$1Cn$$−$$1+$$…$$+n+n$$−$$1Cn$$−$$1=nCn$$−$$1+n+1Cn$$−$$1+$$…$$+2n$$−$$1Cn$$−$$1=$$ nCn−$$1+nCn+n+1Cn$$−$$1+$$…$$+2n$$−$$1Cn$$−$$1$$−$$nCn=$$ $$n+1Cn+n+1Cn$$−$$1+$$…$$+2n$$−$$1Cn$$−$$1$$−$$nCn-$$ $$-$$ $$-$$ $$-$$ $$-$$ $$-$$ $$-$$ $$-$$ $$-$$ $$-$$ $$-$$ $$-$$ $$-$$ $$-$$ $$-$$ $$-$$ $$-$$ $$-$$ $$-$$ $$-$$ $$-$$ $$-$$ $$-$$ $$-$$ $$-$$ $$-$$ $$-$$ $$-$$ $$-$$ $$-$$ $$-$$ $$-$$ $$-$$ $$--$$ $$-$$ $$-$$ $$-$$ $$-$$ $$-$$ $$-$$ $$-$$ $$-$$ $$-$$ $$-$$ $$-$$ $$-$$ $$-$$ $$-$$ $$-$$ $$-$$ $$-$$ $$-$$ $$-$$ $$-$$ $$-$$ $$-$$ $$-$$ $$-$$ $$-$$ $$-$$ $$-$$ $$-$$ $$-$$ $$-$$ $$-$$ $$-$$ $$-=$$ $$2n$$−$$1Cn+2n$$−$$1Cn$$−$$1$$−$$nCn=2nCn$$−nCn∴ $$W=2nCn$$−$$1$$

The number of non-negative solutions of $x_{1} + x_{2} + x_{3} +, \dots, + x_{n} \leq n$ (where n is positive integer) is

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The number of non-negative solutions of x1+x2+x3+,…,+xn≤n (where n is positive integer) is

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The number of non-negative solutions of $x_{1} + x_{2} + x_{3} +, \dots, + x_{n} \leq n$ (where n is positive integer) is