Question

Let S denote the 10-element set {a,b,c,d,e,f,g,h,i,j}. How many ways can we construct a subset of S of size 7 ? 120 How many ways can we construct a subset of S of size 7 containing the element j? 84 How many ways can we construct a subset of S of size 7 containing i but not j ? 28 How many ways can we construct a subset of S of size 7 containing h but neither i nor j ? 7 How many ways can we construct a subset of S of size 7 containing g but not h, i or j? 1 Note that every subset of S of size 7 falls into exactly one of the categories described in parts (b) through (e) above. Use that fact to derive a summation formula involving expressions nCr.

Answer 1

(a) How many ways can we construct a subset of S of size 7 ? 120

answer is ¹⁰C₇=10!/(7!*(10-7)!)=10!/(7!*3!)=120

(b) How many ways can we construct a subset of S of size 7 containing the element j? 84

since j is already selected, now we have to select 6 from 9 element set( {a,b,c,d,e,f,g,h,i}

answer will be ⁹C₆= (9!)/(6!*3!)= 84

(c) How many ways can we construct a subset of S of size 7 containing i but not j ? 28

here we donot want j, it means now we have 9 element {a,b,c,d,e,f,g,h,i}

also we want i to be selected so we we now select 6 element from 8 element set {a,b,c,d,e,f,g,h}

answer will be ⁸C₆=8!/(6!*2!)=28

(d) How many ways can we construct a subset of S of size 7 containing h but neither i nor j ? 7

here now we have effective set {a,b,c,d,e,f,g,h} and want to select h compulsory so

answer will be ⁷C₆=7

(e) here effective set is {a,b,c,d,e,f,g}

g to be selected cumpulsory so number of way will be here ⁶C₆=1

How many ways can we construct a subset of S of size 7 containing g but not h, i or j? 1 Note that every subset of S of size 7 falls into exactly one of the categories described in parts (b) through (e) above. Use that fact to derive a summation formula involving expressions nCr.