Question

A person has $20,000 to invest in four possible opportunities. Each opportunity will accept investments in $1,000 units, but only after the required minimum investment has been made.The minimums are $2, $2, $3 and $4 thousand, respectively. How many investment strategies are possible if: (a) An investment must be made in each opportunity? (b) Investment are made in at least three opportunities?

Answer 1

A)

Let x1,x2,x3,x4 be the number of investments made in four opportunity,

s1,s2,s3,s4 be the minimal investment needed to be made in the x1,x2,x3,x4 opportunity resp.

s1 = 2,

s2 = 2,

s3 = 3,

s4 = 4

, s1+s2+s3+s4 = 11.

also,x1 + x2 + x3 + x4 = 20

now, investment left after minimum investment, x1'+x2'+x3'+x4' = 20-11 =9

solving above equation -

The number of different strategies is given by

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B)

atleast 3 means an investment can be made in each of four opportunity or investments must be made in 3 of the 4 opportunities.

an investment can be made in each of four opportunity = 220ways(calculated above)

an investment can be made in each of four opportunity is calculated below

let the investment be made in any three opportunity

then xi1+xi2+xi3=20 and let fourth opportunity left is xi4 and minimum investment of fourth opportunity is Si4

and sum of minimum investment in four opportunity = 11

then investment left = 20 - 11 +Si4 =9+Si4

solving it

ways

now if fouth minimum investment left is 2,2,3,4 respectively ,then this

,

so, total ways=

=572 startegies

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