Question

# In how many ways 20 identical bananas may be divided among 4 persons and if each person is to be given at least one banana?A.10626, 4845B.1771, 969C.2024, 1140D.None of these

Hint: In this question, 20 identical bananas are distributed among 4 persons. This represents the number of ways to partition n identical things in r distinct slots, which can be solved by using the formula ${\text{n + r - }}{{\text{1}}_{{{\text{C}}_{r - 1}}}}$ and to the second question that if each person is given at least one banana, then 4 persons will get each a banana. The remaining 16 bananas are distributed among 4 persons. And with comes back to same formula ${\text{n + r - }}{{\text{1}}_{{{\text{C}}_{r - 1}}}}$

Given, n = 20, r=4
We know that, ‘n’ identical things can be divided in r distinct slots in ${\text{n + r - }}{{\text{1}}_{{{\text{C}}_{r - 1}}}}$ ways.
$\Rightarrow$ $20 + 4 - {1_{{{\text{C}}_{4 - 1}}}}$ = ${23_{{{\text{C}}_3}}}$
Since, ${{\text{n}}_{{{\text{C}}_r}}}$ = $\dfrac{{{\text{n}}!}}{{\left( {{\text{n - r}}} \right)! \times {\text{r!}}}}$
Therefore, ${23_{{{\text{C}}_3}}} = \dfrac{{23!}}{{\left( {23 - 3} \right)! \times 3!}} = \dfrac{{23!}}{{20! \times 3!}} = \dfrac{{23 \times 22 \times 21 \times 20!}}{{20! \times 3!}} = \dfrac{{23 \times 22 \times 21}}{{3 \times 2}} = 23 \times 11 \times 7 = 1771$
20 Identical bananas can be divided can be distributed among 4 persons in 1771 ways
And now each person gets at least one banana,
$\Rightarrow 20 - 4 = 16$
Therefore, here n=16 and r=4
We know that, ‘n’ identical things can be divided in r distinct slots in ${\text{n + r - }}{{\text{1}}_{{{\text{C}}_{r - 1}}}}$ ways.
$\Rightarrow$ $16 + 4 - {1_{{{\text{C}}_{4 - 1}}}}$ = ${19_{{{\text{C}}_3}}}$
Since, ${{\text{n}}_{{{\text{C}}_r}}}$ = $\dfrac{{{\text{n}}!}}{{\left( {{\text{n - r}}} \right)! \times {\text{r!}}}}$
Therefore, ${19_{{{\text{C}}_3}}} = \dfrac{{19!}}{{\left( {19 - 3} \right)! \times 3!}} = \dfrac{{19!}}{{16! \times 3!}} = \dfrac{{19 \times 18 \times 17 \times 16!}}{{16! \times 3!}} = \dfrac{{19 \times 18 \times 17}}{{3 \times 2}} = 19 \times 3 \times 17 = 969$
20 Identical bananas can be divided and can be distributed among 4 persons with each person being given at least one banana in 969 ways.

Note: We can solve the second question using, the number of ways to partition n identical things in r distinct slots so that each slot gets at least 1 is given by ${\text{n - }}{{\text{1}}_{{{\text{C}}_{r - 1}}}}$ and Number of ways to partition n distinct things in r distinct slots is given by ${{\text{r}}^{\text{n}}}$