There are 5 different pairs of gloves, where left and right are distinguishable. Select 4 of the 10 gloves. (a) How many are there to select 2 pairs of gloves? (b) How many ways are there to select 4 gloves out of the 10 such that 2 of the 4 make a pair. (a pair consists of any right glove and left glove.)

(a) How many are there to select 2 pairs of gloves?

10 ways

(b) How many ways are there to select 4 gloves out of the 10 such that 2 of the 4 make a pair. (a pair consists of any right glove and left glove.)

130 ways

Step-by-step explanation:

For the above questions we apply that combination formula

(a) How many are there to select 2 pairs of gloves?

There are 5 pairs of gloves according to the question above, hence:

5C2 = 5!/2! × (5 - 2)!

= 5!/2! × 3!

= 5 × 4 × 3 × 2 × 1/2 × 1 × 3 × 2 × 1

= 10 ways.

Therefore, there are 10 ways to select 2 pairs of gloves

(b) How many ways are there to select 4 gloves out of the 10 such that 2 of the 4 make a pair. (a pair consists of any right glove and left glove.)

i) A way to select 4 gloves out of 10 gloves =

10C4 = 10!/4! ×(10 - 4)!

= 10!/ 4! × 6!

= 210 ways

ii) In order for 2 of the 4 gloves selected to be a pair, note that we have 5 pairs of gloves hence:

5 × 2⁴

= 80 ways.

Therefore, the number of ways which we can select 4 gloves out of the 10 such that 2 of the 4 make a pair. (a pair consists of any right glove and left glove.) = 210 ways - 80 ways

= 130 ways