Question

In how many ways can Rs. 16 be divided into 4 people when none of them get less than Rs. 3.

• A. 70
• B. 35
• C. 64
• D. 192

Answer 1

The correct option is B

35

x + y + z + w = 16

x, y, z and w should get minimum 3 and maximum we can allot for x, y, z and w = 16 - 9 = 7

Where $3\le p_1,p_2,p_3,p_4\le 7$

Number of ways = Coefficient of $x^{16}$

$(x^3+x^4+x^5+.........+x^7)(x^3+x^4+x^5+............+x^7)(x^3+x^4+x^5+..........+x^7)(x^3+x^4+x^5+...........+x^7)$

= Coefficient of $x^{16}$ in $(x^3+x^4+x^5+..........+x^7{)}^4$

= Coefficient of $x^{16}$ in $x^{12}$$(1+x+x^2+..........+x^4{)}^4$

= Coefficient of $x^{16}$ in $x^{12}$$(1-x^5{)}^4$$(1-x{)}^{-4}$

= Coefficient of $x^4$ in $(1-x^5{)}^4(1-x{)}^{-4}$

= Coefficient of $x^4$ in (1 - 4$x^5$ + ..........)

$[1+4x+.............+\frac{(r+1)(r+2)(r+3)}{3!}{x}^r]$

=$\frac{(4+1)(4+2)(4+3)}{3!}=35.$

Aliter: A + B + C + D = 16

We have to give at least 3 so, give 3 rupees to each one at the beginning

A(3) + B(3) + C(3) + D(3) = 16

A + B + C + D = 4

Remaining 4 rupees can be distributed in ${}^{4+4-1}{C}_{4-1}$ i.e., 35 ways.