Question

Find the number of positive integral solutions of $abc = 45$.

Answer 1

Here we need to use the given positive integral solutions and we also need to find out the answers with help of the formula of permutation and combination of positive integral solutions then we can get the answer.

Complete step-by-step answer:
Given positive integral solutions is
$abc = 45$
We can rewrite above solution as
\Rightarrow$$$a \times b \times c = 45$$ Here Prime number form is \Rightarrow$$$a \times b \times c = {3^2} \times {5^1}$$................(1)
After isolating a, b and c we can write ,

a = {3^{{a_1}}} \times {5^{{a_2}}} \\\ b = {3^{{b_1}}} \times {5^{{b_2}}} \\\ and,c = {3^{{c_1}}} \times {5^{{c_2}}} \ $$ Therefore , product of the form in equation (1) is $\Rightarrow$$${3^{\left( {{a_1} + {b_1} + {c_1}} \right)}} \times {5^{\left( {{a_2} + {b_2} + {c_2}} \right)}} = {3^2} \times {5^1}$$ Comparing the equation $\Rightarrow$$${a_1} + {b_1} + {c_1} = 2$$.....................(2) And $${a_2} + {b_2} + {c_2} = 1$$....................(3) We know that Number of positive integral solutions of equations $${a_1} + {b_2} + ........ + {a_n} = r$$ Number of ways in which $$r$$ identical balls can be each distributed into $$n$$ distinct boxes where each box must contain at least one ball $$ie,$$$$^{n + r - 1}{C_{r - 1}}$$ Now from equation (2) , we have, $${a_1} + {b_1} + {c_1} = 2$$ Or we can write $$ = ^{2 + 3 - 1}{C_{3 - 1}}$$ because the total number n=2 and the possible outcome r is 3. $${ = ^4}{C_2}$$ After solving $$ = \dfrac{{4 \times 3}}{{2 \times 1}}$$ $$ = 6$$ Now from equation (3) $${a_2} + {b_2} + {c_2} = 1$$ we can rewrite it as $$ = ^{1 + 3 - 1}{C_{3 - 1}}$$ because the total number n=1 and the possible outcome r is 3. $${ = ^3}{C_2}$$ After solving $$ = \dfrac{{3 \times 2}}{{2 \times 1}}$$ $$ = 3$$ We have positive integral solution So , required number of positive integral solution $$ = 6 \times 3$$ $$ = 18$$ **Therefore , the number of positive integral solutions is $$18$$.** **Note:** A permutation is defined as the ordering of a set of objects. Like arranging five people in a line is equivalent to finding permutations of five objects. Also the determinant is often defined using permutations. Combination is defined as a selection of items from a collection, such that the order of selection does not matter . so the difference between permutations and combinations is ordering. In permutations we care about the order of the elements, whereas we don’t care about the order with combinations.