Question

A box contains 20 balls. in how many ways can 8 balls be selected if each ball can be repeated any number of times?

Answer 1

This question is a combination problem since the number of times a ball is repeated is not specific.

The number of combinations of n objects taken r at a time is determined by :

C(n, r) = n! / {(n-r)! r!}

! = means factorial and can be obtained from the calculator or can be done manually.

Example : n! Means : (n × n-1 × n-2 ×...... ×1)

In this case n =20 , r =8

Substituting in the formulae we have :

20!/{(20-8)!8!} = 20!/(12!8!)

This equals to 125970 times.

Answer 2

Solution :-

Each combination should have 8 balls out of 20 balls.

Each of the 20 balls may be included in any of the combinations any number of times.

So, it is a case of combination with repetition.

Required number of combinations = (n+r-1)Cr

Here n = 20 and r = 8

⇒ (20 + 8 - 1)C₈

⇒ 27C₈

⇒ (27 × 26 × 25 × 24 × 23 × 22 × 21 × 20)/(8 × 7 × 6 × 5 × 4 × 3 × 2 × 1)

⇒ 89513424000/40320

= 2220075 ways

Answer.