How many integers must you select {without replacement) from the range [1,101] b

Question

How many integers must you select {without replacement) from the range [1,101] before you can ensure: at least one odd number was selected? at least one even number was selected? a perfect square was selected? there exists a pair in your selected set that sum to an even number? (For each of the following be sure to show your work.) Use the binomial theorem to: give an expansion of (x + y)^5 find the coefficients of (x + y) in (x + y)^15 find the coefficient of x^7,y^8 in (5x - 2y)^22 find the coefficient of y^2 in (1 - y)^51

Explanation / Answer

2.

a)

There are 51 odd and 50 even numbers here.

Hence, we can draw 50 even, then finally pick an odd.

Hence, we should pick 51. [ANSWER, 51]

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b)

There are 51 odd and 50 even numbers here.

Hence, we can draw 51 odd, then finally pick an even.

Hence, we should pick 52. [ANSWER, 52]

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c)

There are 10 perfect squares and 91 non-perfect squares here.

Hence, we can draw the 91 non-perfect square, then finally draw a perfect square.

Hence, we pick 92. [ANSWER, 92]

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d)

Note that 2 odd numbers or 2 even numbers add up to an even number.

Hence, we can pick 1 odd, then one even, but we cannot avoid 2 oods or 2 evens on the third pick.

Hence, we pick 3. [ANSWER, 3]

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