NEED HELP BAD!!! TOUGH Questions, TEST TODAY JUST REALIZED

[splq48]

1) A grocery store makes up fruit baskets using as many as four apples, three peras, adn for oranges. A basket must contain at least on piece of fruit. How many different fruit selections are possible?

2)A 10-letter word consists of 4 A's and 6 B's. HOw many different words are possible if no two A's can be next to eachother? (hint start with BBBBBB and decide where to put the A's)

3) A computer manufacturer has 50 distinct microchips to place into a rectangular array that is 5 units wide by 10 units long:
a)In how many ways can the chips be arranged?
b)Ten of the chips control special functions. How many arrangements are possible if these occupy the 1st column
c) Find the number of arrangements having no special-function chips in the 1st column.

 

[tkhunny]

Those are very nice problem statements. Waht have you done to solve them?

 

[splq48]

well for number 1 i started to think that it is possible just to b able to have like 4! x 3! x 4!, which i am pretty sure is completely wrong....I was thinking that it is possible for them to have 4 apples, 3 pears, and then 1 orange, then go the other ways too.

for number 2 i started to think that you can't put any of the A's next to
each other. So i was going to count how many places there are for A's in the sequence and how many A's you need to place in the sequence, but i got hung up in doing so and now i cant seem to get an answer.

For number 3 i am completly lost and i dont kon where to head. I am just not any good at these types of things.

 

[splq48]

anybody have any ideas on where to go and how do figure these out because i am at a loss

 

[splq48]

i think number 1 is 99.....

 

[tkhunny]

Re: NEED HELP BAD!!! TOUGH Questions, TEST TODAY JUST REALIZ

1) A grocery store makes up fruit baskets using as many as four apples, three pears, and four oranges. A basket must contain at least on piece of fruit. How many different fruit selections are possible?

Your problem here is the variable-sized basket. It has anywhere from one piece of fruit up to 11 pieces of fruit. You'll just have to count them, I think.

1-Basket

1 Apple
1 Pear
1 Orange

3 so far

2-Baskets

2 Apples
1 Apple + 1 Pear
2 Pears
1 Pear + 1 Orange
2 Oranges
1 Apple + 1 Orange

6 more. We're up to 9.

I get nine more for 3-Baskets. After that it's a little different, since there is a pear shortage. After that, other options start to drop off.

How did you get 99?

 

[pka]

For #1, expand this expression:
\(\displaystyle \left( {\sum\limits_{k = 0}^4 {x^k } } \right)^2 \left( {\sum\limits_{k = 0}^3 {x^k } } \right) = 1 + 3x + 6x^2 + 10x^3 + 14x^4 + 16x^5 + 16x^6 + 14x^7 + 10x^8 + 6x^9 + 3x^{10} + x^{11}\).
Now the coefficients of the x’s tell us the number of different baskets.
For example, 16x<SUP>5</SUP> means that there are 16 ways to make a five piece basket.
The answer is indeed 99!

For #2, the six B’s will separate the four A’s: ^B^B^B^B^B^B^.
We can put the A’s in any place of ^: Combin(7,4).

For #3, I would have to know if the chips are distinct.