Question

Number of ways in which $13$ identical apples can be distributed among $3$ persons so that no two persons receive equal number of apples and each one can receive none one or more, is

• A. $90$
• B. $84$
• C. $105$
• D. $14$

Answer 1

Answer: (B)

$84$

Total number of ways $={}^{13+3-1}{C}_{3-1}={}^{15}{C}_2$$=105$ ways
But it will also contain the ways when exactly two persons receive equal number of apples & those ways are
0 0 13
1 1 11
$\downarrow$
6 6 1
Number of ways $=7\times 3=21$
Required ways $=105-21=84$ ways