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# In how many ways can two integers m and n, with m > n, be selected fro

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Re: In how many ways can two integers m and n, with m > n, be selected fro [#permalink]
=>

Since the order of $$m$$ and $$n$$ is fixed, we only need to count the number of ways to choose $$2$$ numbers from $$12, 13, …, 32.$$

We have $$21$$ numbers to choose from since $$32 – 12 + 1 = 21.$$

The number of selections is
$$21C2 = \frac{(21*20)}{(1*2)} = 210.$$

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Re: In how many ways can two integers m and n, with m > n, be selected fro [#permalink]
Why not answer 190? there will be 20 combinations where m=n example {12,12}, {13,13}. We need to take out those 20 combinations from 210.
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Re: In how many ways can two integers m and n, with m > n, be selected fro [#permalink]
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PLUTO wrote:
Why not answer 190? there will be 20 combinations where m=n example {12,12}, {13,13}. We need to take out those 20 combinations from 210.

Hey PLUTO

Welcome to GMATClub!

We only choose different integers, as the number of ways of choosing the integers is $$\frac{21*20}{2} = 210$$.

If what you said was possible, the number of ways of choosing the integers are 21*21 = 441
and we will eliminate 21 options - (12,12),(13,13).......(31,31),(32,32). This will bring down
the number of options to 441 - 21 = 420. Now, since (12,21) is the same as selecting (21,12)
we will have to eliminate those options(which are exactly half of the options). Therefore, the
total number of ways of choosing the integers are $$\frac{420}{2} = 210$$

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Re: In how many ways can two integers m and n, with m > n, be selected fro [#permalink]
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Number of integers between a and b (a & b inclusive) = b-a + 1

So, number of integers b/w 12 & 32 (12 & 32 inclusive) = 32-12 + 1 = 21

We shall use combination here as the order does not matter.

For example b/w 12 and 13 the only choice for m>n is (12,13) and not (13,12) and (12,13).

So, the number of combinations possible = 21 C 2

= (21)(20)/(2)(1) = 210

(option d)

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Re: In how many ways can two integers m and n, with m > n, be selected fro [#permalink]
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Re: In how many ways can two integers m and n, with m > n, be selected fro [#permalink]
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