Darren$$s planning a date for a special someone on Valentine$$s date, but he does not know who to pick. He$$s planning to ask $3$ people out $($first choice, back up$,$ and back back up like the intelligent TA he is$),$ out of the $40$ total TA$$s that asked him out. Each person can only have up to $4$ different colored roses $-$ red, orange, blue, and purple. $25$ TA$$s have red roses, $30$ TA$$s have orange roses, $33$ TA$$s have blue roses, and $35$ TA$$s have purple roses. Being the picky TA he is$,$ Darren wants to pick $3$ TA$$s that have all four colored roses. Prove that there exists at least $3$ TA$$s such that each of the $3$ TA$$s have all $4$ possible colored roses.$[30$ pts$]$ Darren's Date Dilemma

Darren's planning a date for a special someone on Valentine's date, but he does not know who

to pick. He's planning to ask $3$ people out $($first choice, back up$,$ and back back up like the

intelligent TA he is$),$ out of the $40$ total TA's that asked him out. Each person can only have

up to $4$ different colored roses $-$ red, orange, blue, and purple. $25$ TA's have red roses, $30$ TA's

have orange roses, $33$ TA's have blue roses, and $35$ TA's have purple roses. Being the picky TA

he is$,$ Darren wants to pick $3$ TA's that have all four colored roses. Prove that there exists at

least $3$ TA's such that each of the $3$ TA's have all $4$ possible colored roses.