test series(combinatorics)

Question

Q.1) The number of integral solution of equation X1+X2+X3+X4=20 where X1>=3,X2>=1,X3>=0,X4>=5  is...................

can anyone explain the approach to solve this question???

Comments

@Swapnil Naik

brother still not getting this

 if you know 9 items are minimum then for the rest 11 we need to find combinations.

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@himgta

see my answer.it might be useful

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yes 364 is correct answer.

Answer 1

The approach would be to subtract the offset part from the total sum and formulate a new problem as below :

y1 + y2 + y3 + y4 = 11, and y1, y2, y3, y4 >= 0.

Now try to solve this question by finding out the number of ways you can put 3 partitions between 11 similar items. This will give you ways to divide 11 items in 4 parts, where each part will correspond to y1, y2, y3 and y4.

Note that in this particular problem there can be only one non-zero y too. That will happen when all 3 partitions are put together at the end or beginning of these 11 items.

Answer 2

Given that:$X_{1}+X_{2}+X_{3}+X_{4}=20$   where $X_{1}\geq3,X_{2}\geq1,X_{3}\geq0,X_{4}\geq5 $

we can write like this :

$X_{1}+X_{2}+X_{3}+X_{4}=20$-------->$(1)$   where $X_{1}-3\geq0,X_{2}-1\geq0,X_{3}\geq0,X_{4}-5\geq0$

Let $X_{1}-3=Y_{1},X_{2}-1=Y_{2},X_{3}=Y_{3},X_{4}-5=Y_{4}$

      $X_{1}=Y_{1}+3,X_{2}=Y_{2}+1,X_{3}=Y_{3},X_{4}=Y_{4}+5$

Put the value in the equation $(1),$

          $Y_{1}+3+Y_{2}+1+Y_{3}+Y_{4}+5=20$  where $Y_{1}\geq0,Y_{2}\geq0,Y_{3}\geq0,Y_{4}\geq0$

    $\Rightarrow Y_{1}+Y_{2}+Y_{3}+Y_{4}+9=20$   where $Y_{1}\geq0,Y_{2}\geq0,Y_{3}\geq0,Y_{4}\geq0$

  $\Rightarrow Y_{1}+Y_{2}+Y_{3}+Y_{4}=11$------>$(2)$   where $Y_{1}\geq0,Y_{2}\geq0,Y_{3}\geq0,Y_{4}\geq0$

The Number of Integral Solution of Equation $=\binom{n+r-1}{r}$ where $n=$Number of Variables and $r=$Sum of the variables.

Here $n=4$ and $r=11$

                                $\Rightarrow\binom{4+11-1}{11}$

                                $\Rightarrow\binom{14}{11}$

                               $\Rightarrow\binom{14}{3}=364$

Comments

Thanks!